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Chapter 9

Estimation and Confidence Intervals

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This chapter considers several different aspects of sampling. Our focus of study begins with point estimates (a single value) and interval estimates (a range of values). We also learn a method to determine the necessary sample size.

Learning Objectives

9-1 Compute and interpret a point estimate of a population mean

9-2 Compute and interpret a confidence interval for a population mean

9-3 Compute and interpret a confidence interval for a population proportion

9-4 Calculate the required sample size to estimate a population proportion or population mean

9-5 Adjust a confidence interval for finite populations

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Point Estimate (1 of 2)

A point estimate is a single value (statistic) used to estimate a population value (parameter)

POINT ESTIMATE The statistic, computed from sample information, that estimates a population parameter.

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Since it is not feasible to contact all the tourists visiting Barbados, the bureau relies on sample information.

Point Estimate (2 of 2)

Example

Suppose the Bureau of Tourism for Barbados wants an estimate of the mean amount spent by tourists visiting that country. They randomly select 500 tourists as they depart and ask these tourists about their spending while there. The mean amount spent by the sample of 500 tourists serves as an estimate of the unknown population parameter.

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Confidence Intervals (1 of 2)

A confidence interval is a range of values within which the population parameter is expected to occur

CONFIDENCE INTERVAL A range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence.

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There are two situations for computing a confidence interval for a population mean; one when the population standard deviation is known, and another when the population standard deviation is unknown.

Confidence Intervals (2 of 2)

The factors that determine the width of a confidence interval for a mean are

The number of observations in the sample, n

The variability in the population, usually estimated by the sample standard deviation, s

The desired level of confidence

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Level of Confidence,σ Known

To determine the confidence limits when the population standard deviation is known, we use the z distribution

The formula is

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Typically, we know the population standard deviation when we have a long history of collected data. Knowing the population standard deviation allows us to use the standard normal distribution.

Finding a Value of z (1 of 3)

The method for finding z for a 95% confidence interval is

Divide the confidence interval in half, .9500/2 = .4750

Find the value .4750 in the body of the table

Identify the row and column and add the values

The probability of finding a value between 0 and 1.96 is .4750

So the probability of finding a value between +/- 1.96 is .9500

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The steps are the same for a 90% confidence interval. In fact, we can select any value between 0 and 100% and find the corresponding value for z. Use the standard normal table found in Appendix B.

Finding a Value of z (2 of 3)

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
.	.	.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.		.	.	.
.	.	.	.	.	.	.		.	.
1.5	0.4332	0.4345	0.4357	0.4370	0.4382	0.4394	0.4406	0.4418	0.4429	0.4411
1.6	0.4452	0.4463	0.4474	0.4484	0.4495	0.4505	0.4515	0.4525	0.4535	0.4545
1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608	0.4616	0.4625	0.4633
1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686	0.4693	0.4699	0.4706
1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.4750	0.4756	0.4761	0.4767
2.0	0.4772	0.4778	0.4783	0.4788	0.4793	0.4798	0.4803	0.4808	0.4812	0.4817
2.1	0.4821	0.4826	0.4830	0.4834	0.4838	0.4842	0.4846	0.4850	0.4854	0.4857
2.2	0.4861	0.4864	0.4868	0.4871	0.4875	0.4878	0.4881	0.4884	0.4887	0.4890
2.3	0.4893	0.4896	0.4898	0.4901	0.4904	0.4906	0.4909	0.4911	0.4913	0.4916
2.4	0.4918	0.4920	0.4922	0.4925	0.4927	0.4929	0.4931	0.4932	0.4934	0.4936

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Finding a Value of z (3 of 3)

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Level of Confidence, z Example (1 of 2)

The American Management Association is studying the income of store managers in the retail industry. A random sample of 49 managers reveals a sample mean of $45,420. The standard deviation of the population is $2,050.

What is the population mean?

What is a reasonable range of values for the population mean?

How do we interpret these results?

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In this example, using the sample mean to construct the confidence interval results in endpoint values of $44,846 and $45,994. To extend the interpretation of these results, suppose we select many samples of 49 store managers, perhaps several hundred. For each sample, we compute the mean of the sample and then construct a 95% confidence interval. We could expect about 95% of these intervals to contain the population mean. About 5% of the intervals would not contain the population mean. This is due to sampling error and is the risk we assume when we select the level of confidence.

Level of Confidence, z Example (2 of 2)

We do not know the population mean, so we can use the sample mean, $45,420 as our best estimate.

The AMA decides to use a 95% level of confidence, so use equation (9-1),

The confidence interval is from $44,846 and $45,994, the value $574 is called the margin of error.

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Level of Confidence, σ Unknown (1 of 2)

To determine the confidence limits when the population standard deviation is unknown, we use the t distribution

The formula is

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In most sampling situations the population standard deviation is not known. In this example, since we do not know , we cannot use formula (9-1). However, we can use the sample standard deviation, s, as an estimate of and replace the z distribution with the t distribution and use formula (9-2).

Level of Confidence, σ Unknown (2 of 2)

Example

The Dean of the Business College wants to estimate the mean number of hours full-time students work at paying jobs each week. He randomly selects a sample of 30 students and asks them how many hours they worked last week. He can calculate the sample mean, but it is unlikely he would know the population standard deviation required for formula 9-1.

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Characteristics of the t Distribution

The t distribution is a continuous distribution

It is mound-shaped and symmetrical

It is flatter, or more spread out, than the standard normal distribution

There is a family of t distributions, depending on the number of degrees of freedom

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These characteristics of the t distribution are based on the assumption that the population of interest is normal, or nearly normal. As sample size increases, however, the t distribution approaches the standard normal distribution.

Finding a Value of t (1 of 3)

First assume the population is normal

Using Appendix B.5, move across the columns identified for confidence intervals

In the next example, we want to use the 95% level of confidence, so move to that column

Then find df, the degrees of freedom, n-1

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When sample statistics are being used, it is necessary to determine the number of values that are free to vary. Here, the degrees of freedom is found by taking the sample size minus 1; 10 – 1 = 9. See the textbook for a more detailed explanation.

Finding a Value of t (2 of 3)

		Confidence Intervals
	80%	90%	95%	98%	99%
		Level of Significance for One-Tailed Test
df	0.10	0.05	0.025	0.010	0.005
		Level of Significance for Two-Tailed Test
	0.20	0.10	0.05	0.02	0.01
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.883	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169

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Finding a Value of t (3 of 3)

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Level of Confidence, t Example (1 of 2)

A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation 0.09 inch. Construct a 95% confidence interval for the population mean.

Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inch?

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A further interpretation of this example is if we repeated this study 200 times, calculating the 95% confidence interval with each sample’s mean and standard deviation, we expect 190 of the intervals would include the population mean. Ten of the intervals would not include the population mean. The calculations to construct a confidence interval are also available in Excel.

Level of Confidence, t Example (2 of 2)

The endpoints of the confidence interval are 0.256 and 0.384. The margin of error is 0.064. The manufacturer can be reasonably sure (95% confident) that the mean remaining tread depth is between 0.256 and 0.384 inch. Because the value 0.30 is in this interval, it is possible that the mean of the population is 0.30.

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Confidence Intervals for Proportions (1 of 3)

PROPORTION The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest.

A sample proportion, p, is found by x, the number of successes, divided by n, the number of observations

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A proportion is a ratio, fraction, or percent that indicates the part of the sample or population that has a particular characteristic. In earlier examples in this chapter we used ratio scale of measurement; in this section we use the nominal scale of measurement since the outcomes will be limited to two values. In the Southern Tech example, a graduate either entered the job market in a position related to his or her field of study or not. Married men either felt that both partners should earn a living or they did not feel that way. In each example, an observation is classified into one of two mutually exclusive groups. We can refer to the groups in terms of proportions.

Confidence Intervals for Proportions (2 of 3)

Examples

Southern Tech career services reports that 80% of its graduates enter the job market in a position related to their field of study

A recent study of married men between the ages 35 and 50 found that 63% felt that both partners should earn a living

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Confidence Intervals for Proportions (3 of 3)

A population proportion is identified by π

Two requirements

The binomial conditions have been met

The values n and n(1-π) should both be greater than or equal to 5

We construct a confidence interval for a population proportion with the following formula

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The population proportion, , refers to the percent of successes in the population. The binomial conditions discussed in chapter 6 must be met: 1. the sample data are the number of successes in n trials; 2. there are only two possible outcomes; 3. the probability of a success remains the same from one trial to the next; 4. the trials are independent. The second requirement allows us to invoke the central limit theorem and employ the standard normal distribution and use z to complete a confidence interval.

Confidence Interval, π Example (1 of 2)

The union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. At least three-fourths of the BBA membership must approve any merger. A random sample of 2,000 current members reveals 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Can you conclude that the necessary proportion of BBA members favor the merger? Why?

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0.80 is calculated to be the sample proportion and is our estimate of the population proportion. Recall, we use a z value of 1.96 for a 95% level of confidence. The formula result has a lower endpoint greater than .75, so we conclude the merger will likely pass. To interpret; if the poll was conducted 100 times with 100 different samples, we expect the confidence intervals constructed from 95 of the samples to contain the true population proportion. This is the procedure used by polling organizations, television networks, and surveys of public opinion on election night.

Confidence Interval, π Example (2 of 2)

Next, use formula (9-4) to determine the 95% confidence interval,

The endpoints of the confidence interval are .782 and .818, so we conclude the merger will likely pass because the interval estimate includes values greater than 75% of the union membership.

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Determining Sample Size for Means (1 of 2)

There are three factors that determine the sample size when we wish to estimate the mean

The margin of error, E

The desired level of confidence, for example 95%

The variation in the population

The formula to determine the sample size for the mean is

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We can determine an appropriate sample size for estimating both means and proportions. Sample size is a decision we make so that our estimate of a population parameter is a good one. The margin of error is the amount of error we are willing to tolerate in estimating a population parameter. We choose relatively high levels of confidence such as 95% and 99% and we use a z statistic. Often, the population standard deviation is not known; so you may decide to conduct a pilot study, use a comparable study, or use a range-based approach to find this value. See the textbook for details.

Determining Sample Size for Means (2 of 2)

The result is not always a whole number; the usual practice is to round up any fractional result to the next whole number.

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Sample Size to Estimate a Population Mean Example (1 of 2)

A student in public administration wants to estimate the mean monthly earnings of city council members in large cities. She can tolerate a margin of error of $100 in estimating the mean. She would also prefer to report the interval estimate with a 95% level of confidence. The student found a report by the Department of Labor that reported a standard deviation of $1,000. What is the required sample size?

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If the student wishes to increase the level of confidence, for example to 99%, this will require a larger sample; in fact, a sample of 664. Larger samples will increase the cost of the study, so the level of confidence should be considered carefully.

Sample Size to Estimate a Population Mean Example (2 of 2)

The computed value of 384.16 is rounded up to 385. A sample size of 385 is required to meet the specifications.

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Determining Sample Size for Proportions

There are three factors that determine the sample size when we wish to estimate a proportion

The margin of error, E

The desired level of confidence

A value for to calculate the variation in the population

The formula to determine the sample size for a proportion is

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If a reliable value for cannot be determined with a pilot study or found in a comparable study, then a value of .50 can be used for . This results in the largest possible value and will overstate the sample size. Using a larger sample size will not hurt the estimate of the population proportion.

Sample Size for the Population Proportion Example

The student in the previous example also wants to estimate the proportion of cities that have private refuse collectors. The student wants to estimate the population proportion with a margin of error of .10, prefers a level of confidence of 90%, and has no estimate for the population proportion. What is the required sample size?

The student needs a random sample of 68 cities.

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Use a population proportion of .50 in the formula since we do not have an estimate for this value.

Finite Population Adjustment

A population that has a fixed upper bound is finite

For a finite population, the standard error is adjusted by the finite-population correction factor,

This will make your estimate more precise by reducing the standard error and resulting in a smaller range of values in estimating the population mean

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A finite population can be small, like all the students in your statistics class or it can be very large, such as all senior citizens living in Florida.

Finite Population Adjustment Example (1 of 2)

There are 250 families residing in Scandia, Pennsylvania. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75.

What is the population mean? What is the best estimate of the population mean?

Develop a 90% confidence interval for the population mean.

Using the confidence interval, explain why the population mean could be $445. Could the population mean be $425? Why?

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The population is finite; there is a limit to the number of people residing in Scandia. We do not know the population standard deviation so we’ll use the t distribution. Using Appendix B.5, move across the top row to 90% and then down to df row 39, the t value is 1.685. Insert the values in the formula and solve. It is likely that the population mean is more than $431.65 but less than $468.35.

Finite Population Adjustment Example (2 of 2)

We do not know the population mean. The best estimate is $450.

The formula is

The endpoints are $431.65 and $468.35, so the population mean could be $445. It is not likely the population mean is $425 since $425 is not within the confidence interval.

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End of Presentation

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