1. Business Statistics For Contemporary Decision Making 9th Edition
Ken Black
Chapter 8
Statistical Inference: Estimation for Single Populations

2. Learning Objectives
Estimate the population mean with a known population standard deviation using the z statistic, correcting for a finite population if necessary.
Estimate the population mean with an unknown population standard deviation using the t statistic and properties of the t distribution.
Estimate a population proportion using the z statistic.
Use the chi-square distribution to estimate the population variance given the sample variance.
Determine the sample size needed in order to estimate the population mean and population proportion. 2 2

3. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
A point estimate is a statistic taken from a sample that is used to estimate a population parameter.
An interval estimate (confidence interval) is a range of values within which the analyst can declare, with some confidence, the population parameter lies.
A point estimate is only as good as the representativeness of its sample.
Because of variation in sample statistics, estimating a population parameter with an interval estimate is often preferable to using a point estimate. 8 11

4. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
100(1-α)% Confidence Interval to Estimate μ: σ known
𝑥 + 𝑧 α/2 𝜎 𝑛 or
𝑥 − 𝑧 α/2 𝜎 𝑛 ≤𝜇≤ 𝑥 + 𝑧 α/2 𝜎 𝑛
where
α= the area under the normal curve outside the confidence interval area
α/2= the area in one end (tail) of the distribution outside the confidence interval 8 11

5. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Example: A cellular telephone company’s would like to estimate the population monthly mean number of texts in the 18-to-24-year-old age category.
From a sample of 85 bills it is determined that the sample mean is 1300 texts.
Using this sample mean, a confidence interval can be calculated within which the researcher is relatively confident that the actual population mean is located.
Suppose that, from previous studies, the population standard deviation is known to be about 160.
The value of z is chosen based on the desired level of confidence.
Common levels of confidence are 90%, 95%, 98%, and 99%.
There is a trade-offs between sample size, interval width, and level of confidence 8 11

6. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Example, continued.
The researcher chooses a 95% confidence level
α = .05; α/2 = .025, the area in each tail.
Looking up ,4750 (= ) in the z table gives a z value of 1.96.
Since the distribution is symmetric, the lower bound is
Now the confidence interval can be estimated, using the formula. 8 11

7. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Example, continued.
𝑥 𝑧 α/2 𝜎 𝑛
≤ 𝜇 ≤
The research company can conclude with 95% confidence that the population mean number of texts per month for an American in the 18-to-24-years-of-age category is between texts and texts.
34.01 (= 𝑧 α/2 𝜎 𝑛 ) is the margin of error.
The margin of error defines the upper and lower bounds of the confidence interval. 8 11

8. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
What does being 95% confident mean?
In 100 random samples of 85 bills, approximately 95 of the 100 intervals would contain the population mean.
In a 90% confidence interval, only 90 of the 100 intervals would be likely to contain the population mean.
In practice, the researcher usually takes only a single sample, but can be 95% confident that the interval that is found includes the population mean.
The figure shows 20 random samples and the intervals found. 8 11

9. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Finite Correction Factor
As in Chapter 7, if the is taken from a finite population, a finite correction factor may be used to increase the accuracy of the solution.
Confidence Interval to Estimate μ Using the Finite Correction Factor
𝑥 − ɀ 𝑎/2 𝜎 𝑛 𝑁−𝑛 𝑁−1 ≤𝜇≤ 𝑥 + ɀ 𝑎/2 𝜎 𝑛 𝑁−𝑛 𝑁−1 8 11

10. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Example: A study is conducted in a company that employs 800 engineers. A random sample of 50 of these engineers reveals that the average sample age is years. Historically, the population
standard deviation of the age of the company’s engineers is approximately 8 years. Construct a 98% confidence interval to estimate the average age of all the engineers in this company.
34.30− −50 800−1 ≤𝜇≤ −50 800−1
31.75≤𝜇≤36.85
Without the correction factor, the interval would be to 8 11

11. 8.1 Estimating the Population Mean Using the z Statistic (σ Known)
Estimating the Population Mean Using the z Statistic When the Sample Size Is Small
If the population can be assumed to be normally distributed, even if the sample size is small, the z distribution can be used.
Using the Computer to Construct z Confidence Intervals for the Mean
Both Minitab and Excel can be used to construct confidence intervals.
Excel gives the margin of error as well as the interval. 8 11

12. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
If the population standard deviation is unknown and the sample size is small, the z distribution cannot be used.
The t Distribution
𝑡= 𝑥 − 𝜇 𝑠 𝑛
The t distribution actually is a series of distributions because every sample size has a different distribution.
Underlying assumption is that the population is normally distributed.
If this assumption cannot be made, nonparametric techniques (Chapter 17) should be used. 8 11

13. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Robustness
If a statistical technique is relatively insensitive to minor violations in one or more of its underlying assumptions, the technique is said to be robust to that assumption.
The t statistic for estimating a population mean is relatively robust to the assumption that the population is normally distributed.
Some statistical techniques are not robust, and a statistician should be careful to be certain that the assumptions underlying a technique are being met before using it.
A business analyst should always beware of statistical assumptions and the robustness of techniques being used in an analysis. 8 11

14. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Characteristics of the t Distribution
t distributions are symmetric, unimodal, and a family of curves.
The t distributions are flatter in the middle and have more area in their tails than the standard normal distribution.
As the sample size becomes large, the t distribution approaches the z distribution. 8 11

15. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Reading the t Distribution Table
The t distribution table is a compilation of many t distributions, with each line of the table having different degrees of freedom and containing t values for different t distributions.
The degrees of freedom for the t statistic presented in this section are computed by n - 1.
The term degrees of freedom refers to the number of independent observations for a source of variation minus the number of independent parameters estimated in computing the variation.
In this case, one independent parameter, the population mean, μ, is being estimated by x in computing s. 8 11

16. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Example: If a 90% confidence interval is being computed, the total area in the two tails is 10%.
Thus, α is .10 and α/2 is .05.
If the degrees of freedom (df) are 24, the t value is located at the intersection of the df value and the selected α/2 value.
In the excerpt from the t table below, the t value is 8 11

17. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Characteristics of the t Distribution
t distributions are symmetric, unimodal, and a family of curves.
The t distributions are flatter in the middle and have more area in their tails than the standard normal distribution.
As the sample size becomes large, the t distribution approaches the z distribution. 8 11

18. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Confidence Intervals to Estimate the Population Mean Using the t Statistic
𝑥 ± 𝑡 𝑎/2,𝑛−1 𝑠 𝑛
𝑥 − 𝑡 𝑎 2 ,𝑛−1 𝑠 𝑛 ≤𝜇≤ 𝑥 + 𝑡 𝑎/2,𝑛−1 𝑠 𝑛
df = n-1
Example: Suppose a researcher wants to estimate the average amount of comp time accumulated per week for managers in the aerospace industry. He randomly samples 18 managers and measures the amount of extra time they work during a specific week.
The sample mean is hours, and the sample standard deviation is 7.80 hours.
The researcher would like a 90% confidence interval.
Since n < 30, and σ is unknown, use the t distribution. 8 11

19. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Example, continued.
With a sample size of 18, there are n-1 = 17 degrees of freedom.
For a 90% confidence interval, α/2 = .05.
t value = 1.740
𝑥 ± 𝑡 𝑎/2,𝑛−1 𝑠 𝑛
13.56±
10.36≤𝜇≤16.76
The researcher is 90% confident that the average amount of comp time accumulated by a manager per week in this industry is between and hours. 8 11

20. 8.2 Estimating the Population Mean Using the t Statistic (σ Unknown)
Using the Computer to Construct t Confidence Intervals for the Mean
The Excel output includes the mean, the standard error, the sample standard deviation, and the error of the confidence interval, referred to by Excel as the “confidence level.”
The confidence interval must be computed from the sample mean and the confidence level.
Minitab gives the confidence interval endpoints. 8 11

21. 8.3 Estimating the Population Proportion
Confidence Interval to Estimate p
𝑝 −𝑧 𝑝 ∙ 𝑞 𝑛 ≤𝑝≤ 𝑝 + 𝑧 𝑎/2 𝑝 ∙ 𝑞 𝑛
where
𝑝 = sample proportion
𝑞 =1− 𝑝
𝑝= population proportion
𝑛= sample size 8 11

22. 8.4 Estimating the Population Variance
The researcher may be more interested in the population variance than the mean or the proportion.
The sample variance is the point estimate of the population variance.
The ratio of the sample variance ( 𝑠 2 ), multiplied by n − 1, to the population variance ( 𝜎 2 ) follows a chi-square distribution ( 𝜒 2 ).
This distribution is NOT robust to the assumption of that the population is normally distributed and should not be used if that cannot be assumed.
𝝌 𝟐 Formula for Single Variance
𝜒 2 = 𝑛−1 𝑠 2 𝜎 2
df=𝑛−1 8 11

23. 8.4 Estimating the Population Variance
Confidence Interval to Estimate the Population Variance
𝑛−1 𝑠 2 𝜒 𝑎/2 2 ≤ 𝜎 2 ≤ 𝑛−1 𝑠 2 𝜒 1−𝑎/2 2
df=𝑛−1
The 𝜒 2 distribution is not symmetrical, and its shape will vary according to the degrees of freedom. 8 11

24. 8.4 Estimating the Population Variance
Example: Suppose eight purportedly 7-centimeter aluminum cylinders in a sample are measured in diameter, resulting in the following values:
The sample variance is 𝑠 2 = , the point estimate.
For a 90% confidence interval with n-1 = 7 degrees of freedom, the values from the 𝜒 2 table are and 8 11

25. 8.4 Estimating the Population Variance
Example, continued:
𝑛−1 𝑠 2 𝜒 𝑎/2 2 ≤ 𝜎 2 ≤ 𝑛−1 𝑠 2 𝜒 1−𝑎/2 2
7 ( ) ≤ 𝜎 2 ≤ 7 ( )
≤ 𝜎 2 ≤
The confidence interval says that with 90% confidence, the population variance is somewhere between and 8 11

26. 8.5 Estimating Sample Size
In most business research that uses sample statistics to infer about the population, being able to estimate the size of sample necessary to accomplish the purposes of the study is important.
Level of confidence, sampling error, and width of estimation interval are closely tied to sample size.
Because of cost considerations, business researchers do not want to sample any more units or individuals than necessary.
Sample Size When Estimating μ
𝑛= 𝑧 𝑎/2 2 𝜎 2 𝐸 2 = 𝑧 𝑎/2 𝜎 𝐸 2
where E = ( 𝑥 − 𝜇), the margin of error of estimation 8 11

27. 8.5 Estimating Sample Size
Often the population variance can be determined from past studies.
If not, an acceptable estimate is 𝜎= 1 4 (𝑟𝑎𝑛𝑔𝑒), where range is the range of values that the x variable can have.
Example: A researcher wants to estimate the average monthly expenditure on bread by a family in Chicago. She wants to be 90% confident of her results, and she wants the estimate to be within $1.00 of the actual figure (error) and the standard deviation of average monthly bread purchases is $4.00. What is the sample size estimation for this problem?
𝑛= 𝑧 𝑎/2 𝜎 𝐸 2 = (4) =43.30
Results should always be rounded up, so she should take a sample of 44. 8 11

28. 8.5 Estimating Sample Size
Determining Sample Size for Estimating p
𝑛= 𝑧 2 𝑝𝑞 𝐸 2
Similar studies can be used to estimate p; if p is unknown, .5 is usually used.
Example: Hewitt Associates conducted a national survey to determine the extent to which employers are promoting health and fitness among their employees. One of the questions asked was, Does your company offer on-site exercise classes? Suppose it was estimated before the study that no more than 40% of the companies would answer yes. How large a sample would Hewitt Associates have to take in estimating the population proportion to ensure a 98% confidence in the results and to be within .03 of the true population proportion?
𝑛= 𝑧 2 𝑝𝑞 𝐸 2 = (.6) = ≅1448 8 11