Business Statistics Online Test
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bstatsmodule7slides.pptxModule 7 Interval estimators
Master for Business Statistics
Dane McGuckian
Topics
7.1 Interval Estimate of the Population Mean with a Known Population Standard Deviation
7.2 Sample Size Requirements for Estimating the Population Mean
7.3 Interval Estimate of the Population Mean with an Unknown Population Standard Deviation
7.4 Interval Estimate of the Population Proportion
7.5 Sample Size Requirements for Estimating the Population Proportion
7.1
Interval Estimate of the Population Mean with a Known Population Standard Deviation
Interval Estimators
Quantities like the sample mean and the sample standard deviation are called point estimators because they are single values derived from sample data that are used to estimate the value of an unknown population parameter.
The point estimators used in Statistics have some very desirable traits; however, they do not come with a measure of certainty.
In other words, there is no way to determine how close the population parameter is to a value of our point estimate. For this reason, the interval estimator was developed.
An interval estimator is a range of values derived from sample data that has a certain probability of containing the population parameter.
This probability is usually referred to as confidence, and it is the main advantage that interval estimators have over point estimators.
The confidence level for a confidence interval tells us the likelihood that a given interval will contain the target parameter we are trying to estimate.
The Meaning of “Confidence Level”
Interval estimates come with a level of confidence.
The level of confidence is specified by its confidence coefficient – it is the probability (relative frequency) that an interval estimator will enclose the target parameter when the estimator is used repeatedly a very large number of times.
The most common confidence levels are 99%, 98%, 95%, and 90%.
Example: A manufacturer takes a random sample of 40 computer chips from its production line to construct a 95% confidence interval to estimate the true average lifetime of the chip. If the manufacturer formed confidence intervals for every possible sample of 40 chips, 95% of those intervals would contain the population average.
The Meaning of “Confidence Level”
In the previous example, it is important to note that once the manufacturer has constructed a 95% confidence interval, it is no longer acceptable to state that there is a 95% chance that the interval contains the true average lifetime of the computer chip.
Prior to constructing the interval, there was a 95% chance that the random interval limits would contain the true average, but once the process of collecting the sample and constructing the interval is complete, the resulting interval either does or does not contain the true average.
Thus there is a probability of 1 or 0 that the true average is contained within the interval, not a 0.95 probability.
The interval limits are random variables because their values depend upon the results of a random sample of data.
However, once they are calculated from a particular sample, those limits are no longer random variables – they become fixed constants, so speaking about their probability in terms of the confidence level is no longer valid.
The Meaning of “Confidence Level”
In the diagram, there are 8 different
confidence intervals represented. Each
confidence interval was constructed using
a sample of size n, drawn from the same
population, and all of the intervals have a
95% level of confidence. The vertical line
in the diagram indicates where the popula-
tion mean is located.
7 of the intervals capture the population mean, but the second interval does not.
If we looked as a very large number of these intervals, approximately 5% (100% - 95%) of them would fail to include the mean.
All of the others would contain the mean as expected.
Confidence Interval for Estimating a Population Mean (with Sigma Known)
The formula for the confidence interval to estimate the mean consists of two values; a lower limit and an upper limit. Confidence Interval for
where n = the sample size
σ = the population standard deviation
= the sample mean
= the z-score separating an area of in the upper tail of the standard normal curve
The formula is often expressed as:
| Lower Limit | Upper Limit |
Confidence Interval for Estimating a Population Mean (with Sigma Known)
However, the most common way to express the formula for the confidence interval to estimate the mean is
where is called the margin of error.
The Margin of Error
In the formula for the confidence interval to estimate the population mean (with σ known), there is a quantity called the margin of error.
The margin of error is the maximum likely difference observed between the sample mean and the population mean , and it is denoted by E.
The margin of error for the confidence interval to estimate the mean is given by the following formula:
where n = the sample size
σ = the population standard deviation
= the sample mean
= the z-score separating an area of in the upper tail of the standard normal curve
The Margin of Error
The margin of error is what determines the width of the confidence interval.
The width of a confidence interval is given by:
When estimating the mean using a confidence interval, the smaller the margin of error, the better.
Since the confidence interval is designed to contain the mean, a narrow interval gives us a better idea of where the mean is located.
The Importance of a Known Population Standard Deviation for a Confidence Interval
The population standard deviation for a random variable is part of the margin of error formula used to estimate the population mean of that random variable.
The reason for this is that the population standard deviation is needed to determine the precise standard error of the sample mean.
If we do not know the precise standard error of the sample mean, we cannot guarantee the level of confidence specified for the interval.
The standard error for :
The margin of error used to estimate
The critical value is determined by assuming the distribution of the sample mean is normally distributed with an unknown mean of and a known standard error of .
The Importance of a Known Population Standard Deviation for a Confidence Interval
If we do not know , then we cannot know the standard error of the sample mean.
This would prevent us from stating an accurate confidence level for our interval estimate.
For this reason, we should know the population standard deviation when using the following confidence interval formula to estimate :
Formula to estimate (with known):
How to Find Critical Z Values
Find the critical Z value needed to construct a 90% confidence interval: .
From the normal table,
looking up the probability
value of 0.4500, we get
because 0.4500 fell
between 1.64 and 1.65.
The Margin of Error when Sigma is Known
The mean quarterly earnings per share for a sample of 36 stocks is $10.52, and the population standard deviation is $2.50. Calculate the margin of error that would be used when estimating the population mean with a confidence level of 95%.
The margin of error is given by:
Here n = 36, , CL = 0.95, So
Then
The Steps to Create a Confidence Interval when Sigma is Known
Step 1: Gather the sample size, sample mean, population standard deviation, and a confidence level
Step 2: Find the Z critical value
Step 3: Calculate the margin of error
Step 4: Calculate the confidence interval
The Confidence Interval for the Mean when Sigma is Known
A student wants to estimate the average amount of time it takes to commute to campus from her apartment. For a random selection of 32 days, she times her commute. The average commute time for those days was 15.3 minutes. Assume the population standard deviation is 3.5 minutes and form a 98% confidence interval to estimate the true mean commute time for the student.
Here n = 32, , CL = 0.98, So
The margin of error is given by:
The 98% Confidence interval is:
So we are 98% confident that the true mean is between 13.86 minutes and 16.74 minutes.
A Confidence Interval
A logistics company claims the true average price for a gallon of regular, unleaded gas is $3.35. A researcher has recently used sample data to form a 98% confidence interval estimate of the true average price of regular, unleaded gas. The interval is given by $3.26 ± $0.06. Do the results contradict the logistics company’s claim?
The 98% confidence interval in this case is: ($3.26 - $0.06, $3.26 + $0.06) = ($3.20, $3.32) which shows that we are 98% confident that the true average price per gallon of unleaded gas will be between $3.20 and $3.32.
So $3.35 is outside of this interval, the results of the researcher does contradict the logistics company’s claim.
On the other hand, if the company had claimed that the average price per gallon of unleaded gas was $3.31, this would have been included in the interval and the results would not have contradicted the company’s claim.
The Factors Affecting the Margin of Error or Width of Confidence Intervals
The Margin of Error determines how wide our confidence interval will be.
We do not want wide confidence intervals because narrow intervals give us a better idea of where the mean lies on the number line.
There are two ways to reduce the error in a confidence interval: (1) decrease the confidence level, or (2) increase the sample size.
Increasing the sample size is not always possible because of costs and implementation considerations.
The population standard deviation is given with the data and cannot be changed.
7.2
Sample Size Requirements for Estimating the Population Mean
The Formula for Determining the Sample Size Needed to Estimate the Population Mean
We can use the formula for the Margin of Error to derive a formula that will tell us the sample size needed to produce a confidence level that has the particular margin of error and a desired confidence level. That formula is:
The Special Rounding Rule for Sample Size Calculations
Remember than n represents the number of subjects that were measured or surveyed in our study, so it cannot be a decimal or a fraction.
When a decimal occurs, we always want to round up because rounding up will produce less error in our confidence interval while rounding down would produce more error.
Determining the Sample Size Needed to Estimate the Population Mean
A stockbroker on Wall Street wants to estimate the average daily-high price for a stock. What sample size is necessary to form a 99% confidence interval to estimate the mean daily-high within 0.50 dollars? Assume the population standard deviation is known to be 4.59 dollars.
Here CL = 0.99,
So the sample size n is:
, by rounding up
Thus the sample size necessary to form a 99% confidence interval is 560.
7.3
Interval Estimate of the Population Mean with an Unknown Population Standard Deviation
Estimating the Mean when the Population Standard Deviation is Unknown
Often we do not know the population standard deviation (σ) when attempting to estimate the population mean using a confidence interval.
When the population standard deviation is unknown, we must use the sample standard deviation as a substitute
However, since the sample standard deviation is not the same as the true population standard deviation, we cannot u se the z distribution to construct the confidence interval.
When the population standard deviation is unknown, we use the t distribution to form our interval estimate of the mean.
Like the z distribution, the t distribution is a bell-shaped distribution, so we will still need to assume the sample mean has a normal (or approximately normal) distribution to use the t distribution.
Estimating the Mean when the Population Standard Deviation is Unknown
When do we use the t distribution to estimate the population mean?
When the population standard deviation (σ) is unknown, and we can assume the distribution of the sample mean is approximately normally distributed.
The Similarities Between the t and z Distribution
It should be stated at the outset that there is not just one t distribution, but a family of t distributions.
For every different sample size n, (degree of freedom), there is a slightly different corresponding t distribution.
These infinitely many t distributions will be defined by their specific degrees of freedom (n – 1).
The family of t distributions is similar to the standard normal (z) distribution in several important ways
The most basic similarity between the t distributions and the standard normal distribution is the fact that they are continuous distributions.
The shapes of the curves are also similar – both distributions are symmetric and mound-shaped (i.e., bell-shaped).
The family of t distribution curves and the standard normal curve have the same mean – that mean is zero.
The Similarities Between the t and z Distributions
The diagram contains the graph of the standard normal distribution and the t distribution for a sample size of 12
The Differences that Exist Between the t and z Curves
The family of t distributions and the standard normal distribution (z) are similar in three ways: (1) both a continuous distributions, (2) both are bell-shaped distributions, and (3) both have a mean of zero.
The differences that exist between the curves all stem from the fact that they have different standard deviations.
The standard deviation for the standard normal curve is 1, whereas for the family of t distributions, the standard deviation varies, but it is always greater than 1.
For every different sample size n (and degree of freedom n – 1), there is a slightly different standard deviation for the corresponding t distribution.
This is the only thing that differentiates the otherwise identical t curves from each other.
The Differences that Exist Between the t and z Curves
Since all probability distributions must have a total area of one, the different standard deviations affect the overall shape of the curves in a predictive way.
Curves with greater variation (a higher standard deviation) will be flatter on top and more spread out.
This means that there is more area in the tails and less in the center of the distribution.
When the curve has less variation (a smaller standard deviation), it will have more data in the center and small tail areas.
The Differences that Exist Between the t and z Curves
The closer the standard deviation is to 1, the more the t distribution will look like the z distribution.
For the family of t distributions, there is an inverse relationship between sample size (degrees of freedom) and standard deviation.
As the sample size increases, the corresponding t distributions have smaller and smaller standard deviations.
This implies that as n increases, the t distributions become more and more like the standard normal distribution.
The Differences that Exist Between the t and z Curves
In the diagram, the two t distributions are graphed along with the standard normal curve. In comparison to the standard normal curve, you can see that the two t curves are thicker (i.e., have more density) in the tails and have less area at the center. The smaller the sample size the more pronounced these differences are.
Confidence Interval for Estimating a Population Mean (with Sigma Unknown)
The formula for the confidence interval to estimate the mean consists of two values; a lower limit and an upper limit:
Confidence Interval for
where n = the sample size
σ = the population standard deviation
= the sample mean
= the t score separating an area of in the upper tail of the t distribution with degrees of freedom
| Lower Limit | Upper Limit |
Confidence Interval for Estimating a Population Mean (with Sigma Unknown)
The formula is often expressed as:
However, the most common way to express the formula for the confidence interval to estimate the mean is
, where
is called the margin of error.
Find Critical t Values
Assuming the population is approximately normal and sigma is unknown, find the appropriate critical value for a 90% confidence interval with a sample size of 20.
Since the population is approximately normal, sample size is small and sigma is unknown, we need to use the t distribution and hence calculate the t critical value. So here:
CL = 0.90
n = 20
df = n – 1 = 19
So the critical value is: (from the t-table on the next slide)
Find Critical t Values: t Table
Form the Margin of Error when Sigma is Unknown
Find the margin of error for a 98% confidence interval estimate of the population mean when sigma is unknown. The sample size is 15. The standard deviation is 20.1, and the data appear to be normally distributed.
Here:
CL = 0.98; n = 15; df = n – 1 = 14; s = 20.1.
So the critical value is: .
So the margin of error is:
The Steps to Create a Confidence Interval (Sigma is Unknown)
The following are the steps to create a confidence interval when sigma is unknown:
Gather the sample data for the problem, which will include and the confidence interval.
Find the critical value
Calculate the margin of error (E).
Form the interval by subtracting the margin of error from the sample mean and adding the margin of error to the sample mean
Construct a Confidence Interval when Sigma is Unknown
A waiter wants to know the average amount of time it takes a table of guests in his section of the restaurant to “turn” (sit, order, eat, pay, and leave). He times a random selection of 25 tables over several busy nights. For those tables, the average time to turn was 42.1 minutes. The sample standard deviation was 4.7 minutes. Assume the turn times are normally distributed, and form a 90% confidence interval for the true mean time to turn a table in this waiter’s section of the restaurant.
Since the population standard deviation is not given and the distribution of turn times is given to be normal, we use the t distribution.
CL = 0.90; n = 25; df = n – 1 = 24; s = 4.7. So
, and the margin of error is .
So the 90% confidence interval for the true mean time to turn is given by
7.4
Interval Estimate of the Population Proportion
Population Proportion
The term proportion refers to the fraction, ratio or percent of the population having a particular trait of interest.
The symbols for population proportion and sample proportion are ρ (rho) and (p-hat) respectively.
Examples of Population Proportions:
In the United States of America, 16.7% of all babies born have blue eyes
In 2013, 31.7% of the U.S. population, aged 25 or older, held a bachelor’s degree or higher.
85% of 18 to 24 year olds, who were raised by at least one parent having a bachelor’s degree or higher, will attend college.
The Sample Proportion
To calculate the proportion of a sample that has some trait of interest, we divide the number of subjects (or items) that have the trait by the number of subjects (or items) belonging to the sample.
Formula for the sample proportion ():
where
x = the number of subjects (or items) having the trait of interest
n = total number of subjects (or items) sampled
The Sample Proportion
For example, consider the survey results below:
The proportion of students reporting that they earned an A in Business Statistics is given by:
| Number of Survey Participants | Number who earned an A in Business Statistics |
| 215 | 20 |
The Sampling Distribution of the Sample Proportion
Recall that if we randomly select n subjects and x of them have some trait we are interested in, the sample proportion formed from the data is:
where x = the number of subjects having the trait we are interested in.
We use as a point estimate of the population proportion (ρ).
For different samples of size n, a different number (x) of subjects will have the trait of interest.
This means the value of will vary from sample to sample.
If we want to use it to form an interval estimate of the true population proportion (ρ), it is important that we know the sampling distribution of .
The Sampling Distribution of the Sample Proportion
The sampling distribution of
is approximately normally distributed
The expected value (mean) for is the population proportion (ρ).
The standard error for is
can be assumed to be approximately normally distributed when both and .
We can approximate the standard error of as
45
The Sample Size Requirement for Estimating the Population Proportion
When constructing a confidence interval to estimate the population proportion, we can assume is approximately normally distributed if both and .
Example: A large corporation wants to estimate the proportion of its part-time employees that would enroll in the company health insurance plan, if it were made available to them. A survey of 500 randomly selected part-time employees reveals that 285 of them would enroll in the plan.
In this example, the sample size is 500 and the number of employees interested in enrolling in the plan is 285. Using these quantities, we can calculate the sample proportion ().
The Sample Size Requirement for Estimating the Population Proportion
Using the sample proportion as an estimate for the population proportion (ρ), we can check the sample size requirement to ensure the sampling distribution of the sample proportion is approximately normal.
and
Since both of the results above are at least 5, it is appropriate to assume the sampling distribution of the sample proportion is approximately normally distributed.
Formula to Calculate a Confidence Interval for the Population Proportion
The formula to calculate the confidence interval for the population proportion ( is given by
where
is the sample proportion
n is the sample size
is the critical value linked to the confidence interval
Constructing a Confidence Interval for the Population Proportion
An efficiency consultant studied a random selection of 200 e-mails received by company employees, to determine how many were relevant to the recipient. Only 36 of the emails were relevant to their recipients. Form a 95% confidence interval to estimate the true proportion of relevant emails received by a typical employee.
Here n = 200, (as only 36 out of the 200 e-mails were relevant), , CL=0.95, .
So (from the z-table on the following slide) and the margin of error is:
So the 95% confidence interval estimate is
We are thus 95% confident that the true proportion of relevant emailed received by a typical employee lies between 0.127 and 0.233.
Question: Can we say that it seems that less than a quarter of emails are relevant? Yes because the upper limit 0.233 < 0.25.
Constructing a Confidence Interval for the Population Proportion
Interpreting a Confidence Interval for the Population Proportion
The CEO of a logistics company claims that only 5% of its holiday deliveries arrive late. A 98% confidence interval to estimate the proportion of late deliveries produced the following interval: 0.06 to 0.11. Does the interval contradict the CEO’s claim?
According to the confidence interval estimate, the true proportion of late deliveries lies between 6% and 11% with 98% confident.
Since both these numbers are higher than the stated value of 5% (that is, the interval does not contain 5%), the CEO’s claim is contradicted.
7.5
Sample Size Requirements for Estimating the Population Proportion
The Formula for Estimating Sample Size for the Population Proportion
The sample size formula when estimating a population is used to specify the sample size required to guarantee that your confidence interval has a certain margin of error and a certain confidence level.
It is derived by taking the margin of error (E) from the confidence interval formula for estimating the population proportion and solving for n.
The equation is:
Since is unknown, we substitute the value 0.5 in the equation because 0.5(1-0.5) is the maximum, so the value of n obtained with this value of will be guaranteed to be as large as it possibly need to be to cover all possible scenarios.
Calculate the Sample Size Needed to Estimate the Population Proportion
A sales manager at a local car dealership wants to estimate the proportion of used car sales that include an extended warranty. What size sample would be needed to estimate the proportion of extended warranties sold with error of no more than 0.05 and a confidence level of 99%?
Here E = 0.05, CL=0.99, , and we use
We use the formula:
So n = 664, as we always “round up” in case of sample size determination.
