Chapter 8 Estimation of the Mean and Proportion In statistics, we collect samples to know more about a population. If the sample is representative of the population, the sample mean or proportion should be statistically close to the actual population mean or proportion. One statistical method to estimate an unknown population mean or proportion is to create a confidence interval. This chapter will describe how to use the calculator to compute confidence intervals for population means and proportions. Confidence Intervals for Population Means There are two functions used to compute confidence intervals for the population mean µ: ZInterval for when σ is known, and TInterval for when σ is unknown. Both are found by pressing and looking in the TESTS menu. Known Population Standard Deviation σ If you are fortunate enough to know the population standard deviation σ, either from theory or from a pilot study, then you would use a Z-based confidence interval, ZInterval, to estimate the population mean, µ. There are two different options for input with the ZInterval function. If you know the sample statistics, then select Stats at the Inpt prompt. Then enter values for σ,, and n. Finally enter the confidence level as a decimal, and select Calculate. If you have the sample data stored in a list, then select Data at the Inpt prompt. Next enter values for σ, List, and Freq. Finally enter the confidence level as a decimal, and select Calculate.
Example: Textbook Price A publishing company has just published a new college textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 25 such textbooks and collected information on their prices. This information produced a mean of $145 for this sample. It is known that the standard deviation of the prices of all such textbooks is $35. Construct a 90% confidence interval for the mean price of all such college textbooks. We have the population standard deviation, σ, so we will use ZInterval command. We do not have the data itself, so we will select Stats at the Inpt prompt. We will enter 35 for σ, 145 for, 25 for n, and.90 for C-Level. Press and select 7: ZInterval from the TESTS menu. At the Inpt prompt, highlight Stats and press. At the σ prompt, type 35. At the prompt, type 145. At the n prompt, type 25. At the C-Level prompt, type.90. The ZInterval output shows the 90% confidence interval, as well as the sample mean and sample size. With 90% confidence, we believe that the true population mean price is between $133.49 and $156.51. Example: Veterinary Bills A random sample of 10 dog owners in a large city was asked to give the dollar amount of their dog s vet bill. Suppose that the standard deviation of the cost of all vet bills (for dogs) in that city is known to be $25. Construct a 95% confidence interval for the mean vet bill (for dogs). The data are $45, $90, $84, $120, $65, $60, $58, $77, $90, and $100. Press and select 1: Edit from the EDIT menu. Enter the data into L 1. We have the population standard deviation, σ = $25, so we will use ZInterval command. We have data, so we will select Data at the Inpt prompt. We will enter L 1 for List, 1 for Freq, and.90 for C-Level. 2
Press and select 7: ZInterval from the TESTS menu. At the Inpt prompt, highlight Data and press. At the σ prompt, type 25. At the List prompt, enter L1 by pressing selecting the list name from the NAMES menu. At the Freq prompt, type 1. At the C-Level prompt, type.95. The ZInterval output shows the 90% confidence interval, as well as the sample mean, sample standard deviation, and sample size. With 95% confidence, we believe that the true mean vet bill is between 45 and 47.1. and Unknown Population Standard Deviation σ In many statistical applications, the population standard deviation is not known. When this is the case, the (standardized values of the) sample mean has a t-distribution, rather than a normal distribution. Thus, a t-based confidence interval, TInterval, is used to estimate the population mean µ. Remember, an underlying condition for safely constructing a t-interval is that either the population is normal or the sample size is larger than 30. Just as for the ZInterval command, there are two different syntaxes for the TInterval command. If you know the sample statistics, then select Stats at the Inpt prompt. Next enter values for, s, and n. Finally enter the confidence level as a decimal, and select Calculate. If you have the sample data stored in a list, then select Data at the Inpt prompt. Then enter values for List and Freq. Finally enter the confidence level as a decimal, and select Calculate. Example: Household Debt A local orange grove sells oranges at Saturday s Downtown Farmer s Market. They wanted to estimate the average number of oranges sold on a given Saturday. They took a sample of 35 Saturdays and found that the average number of oranges sold for this sample is 256 with a standard deviation of 40. Construct a 99% confidence interval for the population mean µ. 3
We do not have the population standard deviation, σ, so we will use the TInterval function. We do not have the data itself, so we will select Stats at the Inpt prompt. We will enter 256 for, 40 for S x, 35 for n, and.99 for C-Level. Press and select 8:TInterval from the TESTS menu. At the Inpt prompt, highlight Stats and press. At the prompt, type 256. At the S x prompt, type 40. At the n prompt, type 35. At the C-Level prompt, type.99. The TInterval output shows the 99% confidence interval along with the sample mean, sample standard deviation, and sample size. With 99% confidence, we believe that the true population mean number of oranges sold on Saturdays is between 237.55 and 274.45 oranges. Confidence Intervals for Population Proportions The function 1-PropZInt computes Z-based confidence intervals for a population proportion when the sample size is large enough (i.e. when both np and nq are greater than 5). 1-PropZInt is found by pressing STAT and looking at the TESTS menu. To use 1-PropZInt, enter the number of successes at the x prompt, the sample size at the n prompt, and the confidence level at the C-Level prompt. Note: x must be a whole number. If you are finding x by multiplying by n, you will need to round to the nearest whole number. Example: Legal Advice A recent sample of 500 college students revealed that 82% of them owned a graphing calculator. Find a 95% confidence interval for the percentage of all college students who own a graphing calculator. 4
In our sample of 500 college students, there were 82% or 410 successes and 90 failures. We can use the 1-PropZInt function with x = 410, n = 500, and our C-Level set at 0.95. Press and select A: 1-PropZInt from the TESTS menu. At the x prompt, type 410. At the n prompt, type 500. At the C-Level prompt, type.95. The 1-PropZInt output shows the 95% confidence interval, the sample proportion, and the sample size. With 95% confidence, we believe that the true population proportion of college students who own their own graphing calculator is between 0.786 and 0.854. 5