Statistics for Managers 2
5ToGo
bus308_chapter_11.pdf
11
Confidence Intervals
Learning Objectives
After reading this chapter, you should be able to:
• Distinguish between point and interval-estimates of values.
• Calculate confidence intervals for t-tests and regression solutions.
• Explain the factors that affect the width of a confidence interval.
iStockphoto/Thinkstock
tan81004_11_c11_273-294.indd 273 2/22/13 3:44 PM
CHAPTER 11Introduction
Chapter Outline
11.1 A Confidence Interval for a One-Sample t-Test Confidence Intervals and the Significance of the Test The Width of the Interval
11.2 A Confidence Interval for an Independent Samples t-Test An Independent Samples t-Test Example Calculating the Confidence Interval of the Difference
11.3 The Confidence Interval of the Prediction The Standard Error of the Estimate Calculating the Confidence Interval of the Prediction Regarding the Width of the Interval The Excel Confidence Intervals for a Regression Solution
Chapter Summary
Introduction
Early in the book there was a distinction drawn between the statistics that describe sample data and the parameter values that describe the characteristics of populations. The relationship between statistics and parameters is important. The value in calculat- ing many statistics is connected to how well they represent one of the possible values of a related parameter. In fact a good deal of analysis and decision making is based on the understanding that when samples are large and randomly selected, the statistics that describe the sample will also provide useful indicators of what the parameter values will likely be. In the progression from one type of analysis to the next in this book, there have been a number of instances in which a statistic was calculated and used in a procedure because the parameter value was not available. In each situation, the statistic that was cal- culated from the sample data was employed as an estimate of what the related parameter value would likely be for the entire population. For example,
• The sample mean, M, is one of the possible values of the population mean, m. • The standard deviation, s, can serve as an estimate of s. • The sample standard error of the mean (SEM) was used to complete the one-
sample t-test in the place of the population standard error of the mean, sM, which the z-test requires.
• The standard error of the difference, SEd, in the independent samples t-test substituted for the standard deviation of the differences sM12M2 between the means of all possible pairs of samples in a distribution of difference scores.
• Ordinary least squares regression produces a predicted value of the criterion variable (y’) when the actual value of that variable, y, is unknown.
In each case, the statistic is a particular, discrete value that takes the place of the related parameter. The value of the statistic is based on the understanding that it is the best estimate available from whatever data are at hand for the value of the more-difficult-to-determine
tan81004_11_c11_273-294.indd 274 2/22/13 3:44 PM
CHAPTER 11Section 11.1 A Confidence Interval for a One-Sample t-Test
parameter. Usually a relatively large sample that is created by randomly selecting the indi- viduals who make it up will reflect the essential characteristics of the population from which the sample was drawn. But “usually” means that sometimes the sample does not accurately represent the population, and the difficulty is that at the time it may not be clear whether the sample is representative.
Many of the statistical procedures used to this point involve calculating a value that indi- cates the degree of difference or association between groups and then determining whether that value is statistically significant. The calculated values represent what are called point estimates of outcomes; they are discrete numbers that are used to determine when a difference or a relationship reflected in sample data is likely to have occurred by chance. The difficulty is that even when a value is based on the most careful data collection proce-
dures, there is always a risk of sampling error. The point estimate by itself provides no indicator of its accuracy. It is difficult to know how much confidence to have in results that are based on these values.
One way to address the limitations of point esti- mates is to move away from the focus on discrete
values and rely instead on what are called interval estimates of the relevant values. For example, rather than asking whether the sample mean provides an accurate estimate of the mean of the population from which the sample was drawn, the question becomes, “What range of values is likely to capture the true value of the population mean from which a significantly different sample was drawn?” In other words, instead of relying on point estimates to estimate population parameters, a confidence interval provides,
with a specified level of probability, a range of values within which the estimated population parameter is likely to fall. Relying on a range of values to capture the value of interest rather than trying to pinpoint it with a discrete value is why this approach to statis- tical analysis makes reference to “confidence inter- vals.” The emphasis is on the use of “interval esti- mates” rather than on “point estimates.”
It isn’t uncommon to see both point and interval esti- mates used in the same analysis. If a statistical test
produces a statistically significant result, the analysis that provided a point estimate of the value of the population parameter is often followed by a confidence interval for that population parameter. That way one can know (a) what the best estimate of the value is, and (b) how much variation should be allowed around that value in order to have a rea- sonable chance of capturing it. Different statistical procedures involve different kinds of confidence intervals.
11.1 A Confidence Interval for a One-Sample t-Test Suppose that the market development team associated with an electric power plant located in Fresno County, California, wishes to estimate average monthly electricity usage. Cal- culating the mean (M) use of electricity among a randomly selected sample of Fresno
Key Terms: Point estimates are discrete values that are cal- culated for unknown parameter values.
Key Terms: Interval esti- mates are calculated ranges within which an unknown parameter value is likely to occur. Because the interval is established based on a specified level of confidence, it is also called a confidence interval.
tan81004_11_c11_273-294.indd 275 2/22/13 3:44 PM
CHAPTER 11
Formula 11.1 C.I..95 5 6t1SEM2 1 M
residents will probably provide a reasonable estimate of mean use among all members of the county, (m), so long as the sample is of an adequate size and is selected in a way that minimizes sampling error. However, what if the goal is to compare domestic use in Fresno County to domestic use in the country as a whole?
As an example, suppose that the mean electricity bill for domestic users in the country as a whole is $230.835 per month. For Fresno County, the monthly costs for 31 ran- domly selected residences average $245, with a standard deviation of 37.555. Since SEM 5 s/ "n , SEM 5 37.555/ "31 5 6.745. Recall from Chapter 4 that t can be calculated as follows:
t 5 M 2 mM
SEM Remembering that uM and m have the same value,
t 5 245 2 230.835
6.745
t 5 2.100
If the probability of a type I error is set at p 5 .05, then the critical value of t for df 5 30 (n 2 1) 5 2.042 (Table 4.1). Average electricity use by the people in Fresno County is sig- nificantly different than it is for the country as a whole. For a one-sample t-test, a statisti- cally significant result indicates that the mean of the population from which the sample was drawn (Fresno) is different from the population mean to which it was compared (United States); to say it differently, it indicates that there are two different populations involved—one is the population from which the sample was drawn (Fresno), and the other is the population to which the sample was compared (United States). But is the sample mean (M) that prompts a significant t value really an accurate estimate of the population from which the sample was drawn? In the electricity use problem, does M provide a reasonably good estimate of electricity use in the population of all electric- ity users in Fresno County? How heavily can one afford to rely on the “M as a good rep- resentation of m” assumption? Rather than calculat- ing the sample mean (M) and relying on sample size and random selection to make the case that M 5 m, the confidence interval of the population mean provides a range of values, thus the reference to an interval, within which mM, the mean of the popu- lation to which the sample does belong, will occur with a specified level of probability. The confidence interval for a one-sample t is calculated with this formula:
Key Terms: The confidence interval of the popula- tion mean is a range within which the value of an unknown population mean has a specified probability of occurring.
Section 11.1 A Confidence Interval for a One-Sample t-Test
tan81004_11_c11_273-294.indd 276 2/22/13 3:44 PM
CHAPTER 11
Where
C.I..95 5 a .95 confidence interval, or a range of values within which the probability is p 5 .95 that the true value of the population mean, mM, will be included.
t 5 the critical value of t for the degrees of freedom associated with the t-test.
The 6 symbol indicates that the critical value of t from the table is included twice, once as a positive value and a second time as a negative value.
SEM 5 the calculated value of the standard error of the mean. M 5 the sample mean.
As is the case with any type of statistical testing, when calculating confidence intervals the analyst deals in probabilities rather than certainties. The range of values that is a .95 con- fidence interval for a one-sample t-test will capture the mean of the population, which is represented by the sample 19 times out of 20. The other side of that coin for a .95 confi- dence interval is that based on averages, 1 time in 20 the parameter value that is sought will occur outside the interval.
The level of probability for the confidence interval is determined by whatever the prob- ability level was for the original test of statistical significance. Since most statistical testing is conducted at p 5 .05, that is also the most common standard for the confidence interval. The confidence interval calculated after a one-sample t that is statistically significant at p 5 .05 must produce a range of values that will miss the true value of the population mean no more than 5 times in 100. However, confidence intervals are stated in terms of the prob- ability of capturing the particular value rather than the proba- bility of missing that value. So instead of a p 5 .05 confidence interval, it is a p 5 .95 confidence interval. If the t-test had been conducted at p 5 .01, a .99 confidence interval would be the appropriate interval estimate to calculate, and so on.
To calculate the confidence interval for the one-sample t procedure completed above, the process is as follows:
C.I..95 5 6t(SEM) 1 M
C.I..95 5 62.042(6.745) 1 245
C.I..95 5 613.773 1 245
C.I..95 5 258.773, 231.227
Review Question A: What is the probabil- ity that the true value will be outside a .95 confidence interval?
Section 11.1 A Confidence Interval for a One-Sample t-Test
tan81004_11_c11_273-294.indd 277 2/22/13 3:44 PM
CHAPTER 11Section 11.1 A Confidence Interval for a One-Sample t-Test
The result is interpreted this way: With .95 confidence, the mean monthly cost of electric- ity in the population to which the Fresno County people belong is somewhere between $231.23 and $258.77.
The initial question that prompted the t-test could be framed this way: Is monthly electric- ity usage in Fresno County consistent with electricity usage for the same month nation- wide? Because the t value is statistically significant, the answer is “no.” The statistically significant t indicates that the sample probably belongs to a population other than the one to which it was compared. Maybe it is summer, and with temperatures higher in Cali- fornia than elsewhere, California consumers have higher electricity usage than what is characteristic of the country as a whole. For whatever reason, the t-test indicates that the sample belongs to a different population than the population that is the country.
The sample mean used in the t-test is a point estimate of the mean of a population. If the t-test result is not statistically significant, the conclusion is that the sample mean is one of the many samples making up the population of sample means. However, the fact that the t-test result is statistically significant indicates that the sample mean is point estimate of the value of some population mean different from what was indicated for the country as a whole. What the point estimate does not indicate, however, is how well the sample mean estimates the value of the mean of the population to which the sample belongs. The confi- dence interval responds to this absence by providing a way to calculate a range of values within which the population mean will probably occur. The “probably” is the reminder that there is never a certainty of including the value. In this case, there is 95% probability that the mean of the population represented by the sample is somewhere between $231.23 and $258.77.
Note that the range of values in the confidence interval does not include the original pop- ulation mean ($230.835) from the initial t-test. When a sample mean in a one-sample t-test is significantly higher than the population mean, the sample mean (M) is probably not one of the possible values of the population mean, a population with mM 5 230.835 in this case. Because the sample mean was significantly higher than mM, the mean of the popula- tion to which the sample probably does belong will have a value higher than 230.835 in the t-test. So the values in the range that is the confidence interval are all beyond the origi- nal value of mM. If a significant t test involved a sample mean less than mM, the confidence interval would include a range with values all lower than mM.
Confidence Intervals and the Significance of the Test
Confidence intervals are only calculated for significant results. If the value of t is not sta- tistically significant, the confidence interval will include a range of values that includes the original value of mM. This happens because a non-significant t-test result is interpreted to mean that the value of mM from the test is one of the values that could be the mean for the population to which the sample was compared. An example will clarify this.
Suppose sales of frozen yogurt for a franchise average $1,125 per day. For a particularly cold week, the daily sales were as follows:
$974, $1,256, $1,170, $842, $875, $1,056, $1,145
tan81004_11_c11_273-294.indd 278 2/22/13 3:44 PM
CHAPTER 11
Are sales during that week significantly different from average daily sales of $1,125? Ver- ify that:
M 5 1045.429
s 5 155.679
SEM 5 s/"n 5 155.679/"7 5 58.841
t 5 M 2 mM
SEM 5
1045.429 2 1125.0 58.841
5 21.352;
With t.05(6) 5 2.447, the result is not significant.
The sample mean from that non-significant result, with the critical value of t and the esti- mated standard error of the mean, produce this .95 confidence interval:
C.I..95 5 6t(SEM) 1 M 5 62.447(58.841) 1 1045.429 5 1189.413, 901.445
The t-test was not significant, and the resulting confidence interval therefore includes the original population mean ($1,125) as one of the values in the interval. Logically, if the point estimate (M) is not significantly different from the population mean (mM), then maybe the mean of the population represented by the sample does have the same value as the speci- fied population mean. At least it is a possibility that cannot be rejected.
The Width of the Interval
The narrower the confidence interval, the more precise estimate the interval provides for the value of the unknown population mean. Sometimes confidence intervals are so wide that they seem not to be very helpful as indicators of the unknown value, so it makes sense to analyze the factors that affect the width of the interval. There are several factors, includ- ing the level of probability at which the interval is calculated. For the electricity problem, the .95 confidence interval stretched from $231.227 to $258.773, around the $245 sample mean. If the market development team wanted a greater level of certainty about capturing the value of the population mean to which the sample belongs, a C.I..99 could be used, but note what happens as a result. First, verify that the critical t value for p 5 .01 and df 5 30 is 2.750. Now the calculated t value of 2.11 is not statistically significant. Second, to calculate the .99 confidence interval:
C.I..99 5 6t(SEM) 1 M
C.I..99 5 62.750(6.745) 1 245
C.I..99 5 618.549 1 245
C.I..99 5 263.549, 226.451
Section 11.1 A Confidence Interval for a One-Sample t-Test
tan81004_11_c11_273-294.indd 279 2/22/13 3:44 PM
CHAPTER 11Section 11.2 A Confidence Interval for an Independent Samples t-Test
The price for a greater certainty of capturing the true value of the population mean (p 5 .99 rather than p 5 .95) represented by the sample is a wider confidence interval and less precision. Now the market development department is 99% certain that the confi- dence interval includes the population mean, but the interval estimate is much broader and probably less helpful. Note that the interval now includes the population mean of $230.835. Precision and certainty push the confidence interval in opposite directions. One is improved only at the expense of reducing the other.
Just as the confidence interval increased when a .95 confidence interval was used, it could have been reduced by calculating a .90 C.I. interval instead (although the particular table in this book does not provide the critical values for significance testing at p 5 .1). Note that with a C.I.90 the probability of missing the true value of the population mean to which the sample belongs is 1 time in 10. As the level of probability is relaxed, the critical value of t diminishes accordingly, which shrinks the C.I.
The other element in the width of the confidence interval is the standard error of the mean, SEM. Since SEM 5 s/ "n , the value of the standard error of the mean can be reduced by either increasing the sample size (larger n, larger divisor, smaller resulting value), finding a way to reduce the variability in the scores so that the standard deviation is smaller, or both. As it turns out, increasing the sample size usually will serve both purposes. Besides the larger n value, it also usually decreases the s value. Small samples tend to be platykur- tic; they ordinarily have relatively large standard deviations given their ranges. As sample sizes grow, particularly randomly selected samples, overall variability tends to decrease. This is consistent with the central tendency characteristics of normal distributions, as more of the scores selected will tend to occur in the middle of the distribution near the mean. Since the standard deviation measures how much individual values tend to vary from the mean of the group, increasing sample size usually shrinks the standard deviation value.
11.2 A Confidence Interval for an Independent Samples t-Test
A statistically significant independent samples t-test indicates that the two samples represented by M1 and M2 probably did not come from the same population; they came from populations with different means. Stated symbolically, m1 2 m2 ? 0. Consistent with the lan- guage used earlier, the two sample means in the case of the independent t-test are point estimates of their respective population means. The more difference there is between M1 and M2 in a statis- tically significant t-test, the greater the difference there probably is between the means of the two related populations. The confidence interval for the independent samples t-test is called the confidence interval of the difference. It provides a way to estimate the difference between the two population means represented by the sample means in a statistically significant independent samples t-test.
The confidence interval of the difference employs this formula:
Key Terms: The confidence interval of the difference is a range probably contain- ing the difference between two unknown population means.
tan81004_11_c11_273-294.indd 280 2/22/13 3:44 PM
CHAPTER 11Section 11.2 A Confidence Interval for an Independent Samples t-Test
Formula 11.2 C.I..95 5 6t(SEd) 1 (M1 2 M2)
Where
C.I..95 5 a range of values within which the difference between the means of the population represented by the samples will be captured with p 5 .95 t 5 the critical value of t for n1 1 n2 2 2 degrees of freedom (the same number of degrees of freedom as there were for the independent samples t) SEd 5 the calculated value of the standard error of the difference M1, M2 5 the two sample means from the independent t-test
As with the one-sample t-test, a confidence interval of the difference is only calculated if the independent samples t-test is statistically significant. Considering the purpose for the confidence interval makes the reason for this evident. What is at issue in the independent samples t is whether the two samples likely represent populations with the same means, or for practical purposes, whether the two samples belong to the same population. That probability is rejected when the result is significant. If the analysis is not significant and the decision is to “fail to reject,” then the samples may come from populations with the same means. It makes little sense, therefore, to calculate a confidence interval for the dif- ference between the population means if there may be only one population involved.
An Independent Samples t-Test Example
A pharmaceutical manufacturer has received Federal Drug Administration (FDA) approval to market a new drug for people with high cholesterol. In two sales regions, pre- scription sales have been similar. But then at some point, one of the two representatives
receives special training on the drug’s chemistry so that the representative can explain more knowledgeably how the drug works and what the side effects are likely to be. The question is whether the extra training is worth the trouble—do the doc- tors visited by the sales rep who received the extra training write a significantly different number of prescriptions than the doctors the other rep visits? The numbers of prescription orders for the drug by doctors in the two sales regions over a seven-week period following the training are as follows:
Extra training: 13, 10, 14, 17, 16, 12, 15 No extra training: 12, 10, 9, 12, 8, 8, 9
Review Question B: What is the relation- ship between the level of confidence and the width of the confi- dence interval?
tan81004_11_c11_273-294.indd 281 2/22/13 3:44 PM
CHAPTER 11Section 11.2 A Confidence Interval for an Independent Samples t-Test
Verify the following descriptive statistics:
Mean Standard Deviation Standard Error of the Mean
Extra Training 13.857 2.410 .911
No Training 9.714 1.704 .644
Recall that SEd 5 "1SEM2 2 1 SEM22 2 5 "1.9112 1 .6442 2 5 1.116
t 5 M1 2 M2
SEd 5
13.857 2 9.714 1.116
5 3.712; t.051122 5 2.179. Reject H0
The result is statistically significant. As the difference between the sample means sug- gests, the doctors visited by the sales representative with the extra training are writing significantly more prescriptions than the doctors in the other sales district. In terms of the number of prescriptions the doctors write, the two sales reps probably now represent two distinct populations. The sample means provide point estimates of the means of the two populations involved. The confidence interval of the difference provides a different way to contrast the two populations by indicating, with the specified probability, what range of values is likely to capture the magnitude of the difference between the two population means.
Calculating the Confidence Interval of the Difference
Formula 11.2 is:
C.I..95 5 6t(SEd) 1 (M1 2 M2)
Note that although the final term in the formula indicates M1 2 M2, it is the absolute value of the difference between the means that the formula requires. If M2 happens to have a greater value than M1, the result will be a difference that is negative, but whether M1 2 M2 is positive or negative isn’t relevant to the confidence interval. It is the absolute value of M1 2 M2 that is entered in the formula.
All the required values are available from the t-test solution above. Substituting them gives the following for the confidence interval:
tan81004_11_c11_273-294.indd 282 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
C.I..95 5 62.179(1.116) 1 (13.857 2 9.714)
C.I..95 5 6.575, 1.711
a. 2.179 3 1.116 1 (13.857 2 9.714) is the upper bound of the confidence inter- val, and
b. 22.179 3 1.116 1 (13.857 2 9.714) is the lower bound of the confidence interval.
The significant independent samples t-test result that indicates that the two samples prob- ably do not represent populations with the same mean is followed with a confidence inter- val estimating how much difference there is between the means of the two populations. The calculations indicate that with p 5 .95 confidence, the difference will be somewhere from 1.711 to 6.575 prescriptions per week.
Note that the confidence interval does not estimate the means of the two populations. Rather, it estimates the difference between their means. The sample means provide esti- mates of the population means, which is as close as we come to identifying their values without further data collection and analysis.
11.3 The Confidence Interval of the Prediction The first two confidence intervals in this chapter were calculated and interpreted in ref- erence to population means. When t is significant in the one-sample test, the confidence interval is a range of values within which the mean of the population represented by the sample is likely to occur. When t is significant in an independent groups test, the confi- dence interval indicates how much difference there is likely to be between the means of the two populations inferred by the samples.
The confidence interval of the prediction has a different orientation. It is a confidence interval for a regression solution, and because the task in regression is to predict the value of y from x (or from multiple x predictors), what the confidence interval produces is a range of values within which the true value of y will probably occur, given a specific value for x.
Remember that the basis for regression is that when- ever variables are significantly correlated, a number of estimates of the value of y from x will be more accurate than a series of random estimates. While that represents a very important theoretical princi- ple, managers are not usually interested in a series of estimates. For example, given a correlation between the price of some product and the resulting sales vol-
ume, a production manager who is trying to forecast what production will need to be in order to satisfy demand is often not interested in a series of estimates of sales volume (y) for several different prices (xs), but rather in one estimate of y from x. The question is often “If the price is ___, what are sales most likely to be?” The value of the confidence interval is that it provides a way to estimate how precise that one prediction of sales volume is likely to be.
Key Terms: The confidence interval of the prediction is a range of probability contain- ing the true value of a criterion variable.
tan81004_11_c11_273-294.indd 283 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
Small values of the standard error of the estimate indicate relatively little error and inter- vals from y’ 2 1(SEest), to y’ 1 1(SEest) that are relatively narrow. When that occurs, there can be increased confidence in the predicted value of y. Small intervals indicate little error in the prediction. On the other hand, if that range is quite wide, it suggests there could be a good deal of variance between the predicted value of y and its actual value.
Capturing the true value of the criterion variable about 2⁄3 of the time, which is what y’ 6SEest provides for, produces essentially a p 5 .68 confidence interval. The problem is that the range of values that is y’ 61(SEest) would miss the true value of y about
1⁄3 of the time. The probability of not capturing the true predicted value in such a confidence interval is too great to be helpful in most analytical situations. To improve the probability of captur- ing y, the more precise C.I..95 and C.I..99 confidence intervals of the prediction are calcu- lated. The formula for a C.I..95 is:
Formula 11.4 C.I..95 5 6tn22(SEest) 1 y’
Formula 11.3 C.I..68 5 61(SEest) 1 y’
It is important to be cautious about the distinction between specificity and precision. The difficulty is not in predicting the value of y given x, which can be done whenever x and y are significantly correlated. The challenge is in making precise predictions of the value of y from x. Although the least squares regression equation provides a particular value that is the best prediction of the criterion variable from a specific value of x, what consti- tutes the best prediction is relative. The best prediction in some circumstances can be very imprecise. A number of factors conspire against accurate predictions, most notably, weak correlations. The problem that the regression solution does not solve is that the predicted value of y, (y’), yields little information about how precise the prediction is likely to be. It is a shortfall that the confidence interval addresses.
The Standard Error of the Estimate
Recall that in any normal distribution, plus or minus 1 standard deviation from the mean, accounts for about 2⁄3 of the population, 68.26% to be more exact. Recall also from Chapter 9 that the standard error of the estimate (SEest), which estimates prediction errors in regres- sion problem, is by definition a standard deviation value. In fact it is the standard deviation of all possible error scores from an infinite number of predictions of y from x. While that definition has important conceptual value, in practice the standard error of the estimate is in fact an estimated value, as the name suggests. Recall that SEest 5 sy"1 2 r2xy . Because it is a type of standard deviation value, and since 6 one standard deviation from the mean in any normal distribution includes 68% of that distribution, we can say by extension that 61(SEest) from the predicted value of y (y’) point should provide a range of values within which the true value of y will occur about 68% of the time.
tan81004_11_c11_273-294.indd 284 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
Where
C.I..95 5 a .95 confidence interval for the regression solution t 5 the critical value of t for n 2 2 degrees of freedom where n 5 the number of pairs of scores SEest 5 the standard error of the estimate y’ 5 the predicted value for the criterion variable
Calculating the Confidence Interval of the Prediction
A produce importer recognizes a relationship between how long it takes produce to get from shipping docks to the retail market and how much of the order is lost to spoilage because of over-ripening. The data for number of days (x) and the percentage of the order lost to over-ripeness (y) is as follows:
Number of days: 12, 15, 11, 9, 7, 9, 12, 10, 7, 8, 9, 8 Percentage lost: 8, 10, 7, 7, 6, 8, 9, 9, 5, 6, 7, 6
With a dock workers’ strike looming and produce coming into port, the importer wishes to predict what the losses will be if it takes 18 days to get the produce to the retailer.
Verify the following descriptive statistics:
Mean Standard Deviation
Number of Days (x) 9.750 2.379
% Lost (y) 7.333 1.497
rxy 5 .868 and is statistically significant at level p 5 .05
y’ 5 a 1 bx
b 5 rxy(sy/sx)
5 .868(1.497 4 2.379)
5 .546
a 5 My 2 bMx
5 7.333 2 (.546)(9.750)
5 2.010
y’ 5 2.010 1 .546(18)
5 11.838
tan81004_11_c11_273-294.indd 285 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
Based on these data, if it takes 18 days to get the produce to retail outlets, there will be an 11.838% loss due to over-ripening.
Like the values for M in the one sample t, and M1 2 M2 in the independent t, the 11.838 value is a point estimate—a discrete number—indicating a solution value. As with t-tests, the confidence interval of the prediction produces an interval estimate, or a range of num- bers within which the true percentage of spoilage will occur with a stated probability. To calculate a .95 confidence interval, first calculate the standard error of the estimate, which is determined as follows:
SEest 5 sy"1 2 r2xy
5 1.497"1 2 .8682
5 .743
The critical value of t in the confidence interval formula is determined by the degrees of freedom for the problem as well as the level of probability for which the interval is calculated. The degrees of freedom are the same as they are for determining whether the correlation coefficient is statistically significant—the number of pairs of data minus two (n 2 2). The level of probability for the confidence interval is ordinarily determined by the level at which the x, y correlation was found to be significant. If the correlation is statisti- cally significant at p 5 .05, usually a .95 confidence interval is calculated. While this is not a hard-and-fast rule, it is typical.
When testing at p 5 .05, the critical value of t for n 2 2 degrees of freedom is t.05 (10) 5 2.228. So to complete the confidence interval:
C.I..95 5 6tn22(SEest) 1 y’
5 62.228(.743) 1 11.838
5 13.493, 10.183
With .95 confidence, if the produce takes 18 days to get to market, the percentage of spoil- age will be somewhere between 10.183% and 13.493%. The importer is now in a position to determine what the financial impact will be for a labor interruption of a particular duration.
Note that if multiple predictors are present, the calculations for the confidence interval of the prediction are very similar. Simply substitute SEest with SEmest, which is calculated using the multiple regression coefficient.
tan81004_11_c11_273-294.indd 286 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
Regarding the Width of the Interval
The factors that affect the width of confidence intervals for t-test problems were listed earlier in the chapter. Note below that many of the same elements are factors in regression problems:
• There is a tension between the level of confidence and the size of the interval. With a .95 confidence interval, the level of certainty is greater than the .68 interval that was y’ 61(SEest), so the interval must be wider as well; the width of the interval increases with the level of probability. A .99 confidence inter- val, which requires a yet greater probability of including the true value of y, must result in a wider confidence interval. The difference in the width of the .99 C.I. compared to a .95 C.I. is due to the critical value of t from the table. Testing at p 5 .01 involves a larger critical value of t than testing at p 5 .05.
• Recall that the critical values of t rise and fall with the number of degrees of freedom. The degrees of freedom, in turn, are based on sample size. As sample size grows, degrees of freedom increase and the critical value, which is a multiplier for the standard error of the estimate, shrinks. For this reason, analyses involving large samples result in narrower confidence intervals than similar problems with fewer data.
• The standard error of the estimate is directly affected by the strength of the x, y correlation (SEest 5 sy"1 2 r2xy ) and by the variability in the criterion variable, y. Weak correlations and highly variable y data (indicated by large standard deviation values) both swell confidence intervals compared to problems with stronger correlations and more homogeneous y data.
The critical value of t for 10 degrees of freedom if testing at p 5 .01 is rxy 5 .708. Since the days-to-market and percentage of spoilage correlation was rxy 5 .868, the correlation is also statistically significant at p 5 .01. Note the difference in the widths of the .99 and .95 confidence intervals:
C.I..99 5 6tn22(SEest) 1 y’
5 63.169(.743) 1 11.838
5 14.193, 9.483
The difference between the widths of these two intervals is illustrated in Figure 11.1.
tan81004_11_c11_273-294.indd 287 2/22/13 3:44 PM
CHAPTER 11Section 11.3 The Confidence Interval of the Prediction
Figure 11.1: Comparing .95 and .99 confidence intervals
The diamonds indicate the (approximate) data points in the graph. The dark blue diagonal line is the regression line, which is based on values for the intercept (a 5 2.010) and the slope (b 5 .546). The value of the predictor variable for the problem was x 5 18. From that value on the horizontal axis vertically to the regression line and then horizontally left back to the y axis provides the prediction for the percentage of the produce that will be lost, the value of y’, which was 11.838%. The blue and brown brackets represent the .95 and .99 confidence intervals, respectively. Both intervals center around y’ 5 11.838; their different widths reflect the different prob- abilities (p 5 .95 or p 5 .99) of capturing the true value of y.
The Excel Confidence Intervals for a Regression Solution
If the regression problem is completed in Excel, the output provides many of the same val- ues that were computed longhand. The confidence intervals, however, are different from what has been done here. If the problem for the number of days delayed and the percent- age of produce lost is done on Excel, the results are those in Figure 11.2.
Number of Days to Retailers (x)
P e rc
e n ta
g e L
o st
( y)
18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16 18
C.I. .99
C.I. .95
y = 11.838
Review Question C: What is the relation- ship between sample size and the width of the confidence interval?
tan81004_11_c11_273-294.indd 288 2/22/13 3:44 PM
CHAPTER 11Chapter Summary
Figure 11.2: The Excel regression output
Rather than calculating a confidence interval for the prediction, y’, what Excel does is produce confidence interval for the a and b values from the regression equation. So rather than answering the question
With p 5 .95, what will be the percentage of product loss if the shipment is delayed by 18 days?
the confidence interval answers this question:
With p 5 .95, what are the true values of the intercept and the regression coef- ficient (slope) values in the regression equation?
The answers to those questions are the values in the last two columns in the final table.
Chapter Summary
When certain necessary conditions prevail, descriptive statistics are useful indicators of population characteristics—what is accessible can be used to describe what is unknown and inaccessible. This is important in many kinds of business analyses. The insurance adjustor may not know what the cost of repairs has been for every “fender- bender” involving particular vehicles, but having the cost data for a representative sample of those repairs can enable a reasonable estimate of the repair cost for one such accident.
0.867669
df
ANOVA
MS F Significance
F
Multiple R
Regression Statistics
Summary Output
R Square
Adjusted R
Square
Standard Error
Observations
Regression
Residual
Total
Intercept
Days
0.752849
0.728134
0.780794
12
1
10
18.57028 30.46113 0.000254853
0.609639
2.008032
0.546185
0.990851851 2.026572
5.519161
0.070206 –0.19972337 4.2157876
0.76668480.3256846420.0002550.09896156
11
18.57028112
6.096385542
24.66666667
Standard Error
t Stat P-value Lower 95% Upper 95%Coefficients
SS
tan81004_11_c11_273-294.indd 289 2/22/13 3:44 PM
CHAPTER 11Chapter Summary
The more the sample resembles the population, the more a particular case drawn from that sample is likely to reflect what occurs more generally in the population. Sometimes, because samples tend to be a good deal more accessible than populations, analysts use the characteristics of the sample to understand those of the population, a sequence that defines inferential statistics.
The “fly in the ointment” is sampling error. Even with random selection and relatively large samples, sampling error can still be present, and the sample may be unlike the popu- lation. This possibility increases when circumstances compel the use of small samples that are derived from nonrandom sampling—convenience samples, for example. When, for whatever reason, samples are not representative, the point estimate that is the calculated statistic may be a poor estimate of the population parameter.
One way to understand confidence intervals is to view them as an indicator of the point estimate’s fidelity. Since confidence intervals produce a range of values within which the true value of the related parameter occurs with a stated probability, narrow confidence intervals indicate point estimates that are probably good indicators of the inferred param- eter (Objective 1). A narrow confidence interval for a one-sample t-test suggests that M is a reasonably good indicator of the value of the population mean to which the sample belongs. A narrow range of values for the confidence interval of the prediction indicates that y’ is probably a good estimate of what the value of y would be if it was known rather than predicted from the sample data (Objective 2).
While determining what constitutes a large or small confidence interval involves at least some element of subjectivity, identifying the factors that affect the size of the interval is straightforward. Each confidence interval includes a statistic that measures data variabil- ity. For the one-sample t it is the standard error of the mean (SEM). For the independent samples t-test it is the standard error of the difference (SEd). For the regression solution, the measure of data variability is the standard error of the estimate (SEest) or the standard error of the multiple estimate (SEmest). In each case these values are used as multipliers in the confidence interval formulas. The implication is that more data variability translates into broader confidence intervals.
Often, greater or lesser data variability can be predicted. Smaller groups, for example, tend to have larger standard deviations—more data variability—than larger groups. Con- fidence intervals based on small groups tend to be wider than those associated with larger groups. There is an important consistency here: Since small samples have a more limited opportunity to emulate the characteristics of populations, it makes sense that confidence intervals based on small groups entail more risk of missing the target value, and they have to be wider to accommodate that risk.
Small samples are associated with larger confidence intervals for another reason. Each confidence interval formula employs a value from one of the tables of critical values. Each critical value for a problem is determined by its associated degrees of freedom. Since small samples provide fewer degrees of freedom and correspondingly larger critical values, they result in larger confidence intervals.
tan81004_11_c11_273-294.indd 290 2/22/13 3:44 PM
CHAPTER 11Chapter Formulas
The final factor in the width of the confidence interval is the level of confidence. In the chapter we noted that confidence and precision are at odds. The most precise (narrowest) intervals involve the greatest risk of missing the parameter value. As the willingness to “miss” the parameter is reduced, the width of the interval must increase so that the prob- ability of including the actual value likewise increases (Objective 3).
Several confidence intervals in statistical analysis haven’t been addressed in this chapter. Those that are here are some of the more useful, however, and some of the easier to calcu- late and understand. Other confidence intervals that the business analyst may encounter are going to be variations on a common theme: They all involve ranges of values around a calculated statistic within which the true value of some unknown value will probably be present.
Answers to Review Questions
A. A 95 confidence interval has a 5% chance (p 5 .05) of missing the true value for which the confidence interval was calculated.
B. As the level of confidence increases, so must the width of the interval. C. As sample size increases, the critical value used in the calculation of the confi-
dence interval declines, and that shrinks the width of the confidence interval. Larger sample sizes also tend to exhibit less variability, which yields narrower confidence intervals.
Chapter Formulas
Formula 11.1 C.I..95 5 6t(SEM) 1 M The formula for the confidence interval of the population mean.
Formula 11.2 C.I..95 5 6t(SEd) 1 (M1 2 M2) The formula for the confidence interval of the difference.
Formula 11.3 C.I..68 5 61(SEest) 1 y’ The formula for the .68 confidence interval of the prediction.
Formula 11.4 C.I. 5 6tn22(SEest) 1 y’ The formula for the confidence interval of the prediction.
tan81004_11_c11_273-294.indd 291 2/22/13 3:44 PM
CHAPTER 11Management Application Exercises
Management Application Exercises
Unless otherwise stated, use p 5 .05 in all your answers.
1. A hardware retailer has averaged sales of $64,235, with a standard deviation of $5,918, for a 12-month period. The mean monthly sales for all retailers in the chain are $59,844. Are this hardware retailer’s sales significantly different from those of all retailers in the chain at p 5 .05? Are they significantly different at p 5 .01?
2. Calculate a .95 confidence interval for the data in problem 1. Explain your findings in lay terms. What question does the confidence interval answer?
3. Calculate C.I..99 for the data in problem 1. What is the effect of changing the confi- dence level on the width of the interval? Why?
4. There are two groups of eight sales representatives for a pharmaceutical company in two regions of the country. For group 1, the average monthly charges to expense accounts are $387, with s 5 $22.73. For group 2, the average is $344, with s 5 18.19. Is the difference between the two groups statistically significant at p 5 .05? Are they significantly different at p 5 .01?
5. Calculate a C.I..95 for the difference for the item 4 data. Which value(s) represent the point estimate of the difference, and which value(s) indicate(s) the interval estimate of the difference for the item 4 data?
6. What could be done to narrow the interval estimate for the item 4 data?
7. A car dealer is using the number of years customers have owned their vehicles to predict how long it will be before they elect to replace them. The correlation between the two is rxy 5 2.723 (the longer they have owned their present vehicles, the more quickly they are expected to replace them). The other relevant data are as follows for 32 customers:
Mean Standard Deviation
Years owned present vehicle 4.551 1.627
Time until expected replacement 2.259 1.338
8. How long will it be until a customer who has owned the vehicle 6.5 years is likely to replace it?
9. Calculate .95 and .99 confidence intervals for the prediction in item 7.
10. How will a larger standard deviation in the criterion variable affect the width of the confidence intervals in item 8? Why?
11. What does a .95 confidence interval of the prediction for the item 7 data reveal?
tan81004_11_c11_273-294.indd 292 2/22/13 3:44 PM
CHAPTER 11Key Terms
Key Terms
• A point estimate is a discrete number used to estimate the value of a related mea- sure. In inferential statistics, the value of the sample mean may be used as an esti- mate of the population mean’s value; M is a point estimate of m.
• An interval estimate is a range of values (rather than a discrete number, as with the point estimate above) calculated so as to contain some unknown value, with a stated level of probability. Because the level of probability indicates how confident one is that the interval will contain the value of interest, interval estimates are usually known as confidence intervals.
• The confidence interval of the population mean is ordinarily calculated follow- ing a statistically significant t value for a one-sample t-test. This confidence interval produces a range of values within which the mean of the population represented by the sample will occur, with a stated probability.
• The confidence interval of the difference is calculated following a statistically significant independent samples t-test. It produces an interval within which the dif- ference between the populations represented by the two samples will occur, with a stated probability.
• The confidence interval of the prediction is calculated in reference to a regression solution. It produces a range of values within which the true value of the criterion will occur, with a stated probability.
tan81004_11_c11_273-294.indd 293 2/22/13 3:44 PM
tan81004_11_c11_273-294.indd 294 2/22/13 3:44 PM
