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Section 8.3

Section 8.3. Estimating Population Means (   Unknown). Estimating Population Means (  Unknown ). Margin of Error of a Confidence Interval for a Population Mean (  Unknown)

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Section 8.3

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  1. Section 8.3 Estimating Population Means ( Unknown)

  2. Estimating Population Means ( Unknown) Margin of Error of a Confidence Interval for a Population Mean (Unknown) When the population standard deviation is unknown, the sample taken is a simple random sample, and either the sample size is at least 30 or the population distribution is approximately normal, the margin of error of a confidence interval for a population mean is given by

  3. Estimating Population Means ( Unknown) Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) Where is the critical value for the level of confidence, c = 1 − , such that the area under the t-distribution with n − 1 degrees of freedom to the right of is equal to s is the sample standard deviation, and n is the sample size.

  4. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) Dental researchers want to estimate the mean leakage, measured in nanometers (nm), of a new filling material for cavities using a simple random sample of 10 trials. Assuming that the population distribution is approximately normal and the population standard deviation is unknown, find the margin of error for a 95% confidence interval for the population mean given that the sample standard deviation is 15.5 nm.

  5. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) Solution Since we know that the population distribution is approximately normal and the population standard deviation is unknown, we are able to use the t-distribution to calculate the margin of error. The problem tells us the values for s and n (s = 15.5, n = 10), so the only missing value in the calculation of E is Since the level of confidence is 95%, α = 1 - 0.95 = 0.05. Therefore, A sample size of 10 means that there are 9 degrees of freedom, df = 9.

  6. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) To find this value by hand using the t-distribution table, look across the row for 9 degrees of freedom and down the column for an area in one tail of 0.025. This shows a critical t-value of Notice that, using our table, we could also have looked up the area in two tails,  = 0.05, instead of the area in one tail, Both give the same answer.

  7. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.)

  8. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) We can also use a TI-84 Plus calculator to find the critical t-value. Recall that you need to enter the area in the left tail only, so remember to divide  by 2 when using the calculator: • Press and then to go to the DISTR menu. • Choose option 4:invT(. • Enter invT(0.025,9).

  9. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) Notice that the value of t that is returned is negative, Because we want the positive value of t, we can just ignore the negative sign since the t-distribution is symmetric.

  10. Example 8.15: Finding the Margin of Error of a Confidence Interval for a Population Mean (Unknown) (cont.) Substituting these values into the formula for the margin of error, we get the following. Although we are not calculating the endpoints of the confidence interval in this example, we will round the margin of error to six decimal places. So the margin of error for this 95% confidence interval is approximately 11.087262 nm.

  11. Estimating Population Means ( Unknown) Confidence Interval for a Population Mean The confidence interval for a population mean is given by Where is the sample mean, which is the point estimate for the population mean, and E is the margin of error.

  12. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) A marketing company wants to know the mean price of new vehicles sold in an up-and‑coming area of town. Marketing strategists collected data over the past two years from all of the dealerships in the new area of town. From previous studies about new car sales, they believe that the population distribution looks somewhat like the following graph.

  13. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.)

  14. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) The simple random sample of 756 cars has a mean of $27,400 with a standard deviation of $1300. Construct a 95% confidence interval for the mean price of new cars sold in this area. Solution Step 1: Find the point estimate. The point estimate for the population mean is the sample mean, which we are told is $27,400.

  15. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) Step 2: Find the margin of error. Next, we need to calculate the margin of error. From the graph, we know that the population distribution is not guaranteed to be normal, but instead is considered skewed to the right. However, since the sample size, n = 756, is sufficiently large, we use the Student’s t-distribution to calculate the margin of error. We are told that the standard deviation of the sample is $1300, that is, s = 1300, so all that remains is to find the critical t-value.

  16. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) Since our sample is so large, we will use the calculator method instead of the table to find As we saw in the previous example, to find for the t-distribution with df = n- 1 = 756 - 1 = 755 degrees of freedom, enter invT(0.025,755) and find that the critical value we need is Notice that this is the same value that you get if you look in the table of critical t-values for df = 750 with a 95% level of confidence.

  17. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) With such a large sample, there is no difference in the first three decimal places of the critical value. Substituting into the margin of error formula, we have the following.

  18. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) Step 3: Subtract the margin of error from and add the margin of error to the point estimate. The third step is to subtract the margin of error that we just calculated from the point estimate we were given in the problem, and then add the margin of error to the point estimate to get the lower and upper endpoints of the confidence interval.

  19. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) Thus, the 95% confidence interval ranges from $27,307 to $27,493.

  20. Example 8.16: Constructing a Confidence Interval for a Population Mean (Unknown) (cont.) The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below. So with 95% confidence, we can say that the mean price of new cars sold in the area is between $27,307 and $27,493.

  21. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) A student records the repair costs for 20 randomly selected computers from a local repair shop where he works. A sample mean of $216.53 and standard deviation of $15.86 are subsequently computed. Assume that the population distribution is approximately normal and  is unknown. a. Determine the 98% confidence interval for the mean repair cost for all computers repaired at the local shop by first calculating the margin of error, E. b. Use a TI-83/84 Plus calculator to determine the 98% confidence interval from the given statistics.

  22. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Solution Because is unknown, we need to ensure that either the sample size is large enough (n ≥ 30) or the population is normally distributed in order to use the t-distribution for the confidence interval. We are given the assumption of normality, so we can proceed. a. Step 1: Find the point estimate. The point estimate for the population mean is the sample mean, which we are told is $216.53.

  23. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Step 2: Find the margin of error. Calculating the margin of error first requires that we find The level of confidence is c = 0.98, so There are 20 computers in the sample, so df = 19. Using either the table of critical t-values or technology, we find that

  24. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Next, substitute these values into the formula for the margin of error.

  25. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Step 3: Subtract the margin of error from and add the margin of error to the point estimate. Subtracting the margin of error that we just calculated from the point estimate we were given in the problem and then adding the margin of error to the point estimate gives us the following endpoints of the confidence interval.

  26. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Thus, the 98% confidence interval ranges from $207.53 to $225.53. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below.

  27. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Therefore, the student can be 98% confident that the mean repair cost for all computers repaired at the local shop is between $207.53 and $225.53.

  28. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) b. Now we’ll show you how to do the same interval calculation using a TI-83/84 Plus calculator. • Press . • Scroll over and choose TESTS. • Choose option 8:TInterval. • Choose the Stats option because we were given sample statistics.

  29. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) We’ll need to enter the following sample statistics, as shown in the calculator screenshot in the margin. • Sample mean, Ë:216.53 • Sample standard deviation, Sx:15.86 • Sample size, n:20 • Level of confidence, C-Level:.98 After highlighting Calculate and pressing , we get the results shown in the second screenshot.

  30. Example 8.17: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.) Notice that the last digits of the endpoints in the interval given by the calculator, (207.52, 225.54), are different from those in our hand-calculated interval. This is because we used a rounded value of to find the margin of error in our first method of calculation.

  31. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data Given the following sample data from a study on the average amount of water used per day by members of a household while brushing their teeth, calculate the 99% confidence interval for the population mean using a TI-83/84 Plus calculator. Assume that the sample used in the study was a simple random sample.

  32. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data (cont.)

  33. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data (cont.) Solution Since we are not told any population parameters for the study, we cannot assume that sis known or that the population distribution is approximately normal. However, since the sample size (n = 40) is large enough (n ≥ 30), we can use the t-distribution to construct a confidence interval for the population mean.

  34. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data (cont.) To begin with, since we are given the raw data and not the sample statistics, we need to enter the data in the calculator list. Recall, to enter data in a TI-83/84 Plus calculator, press , choose EDIT, select 1:Edit, and then enter the data in L1. (Remember to clear the list before entering the data.)

  35. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data (cont.) Once the data are entered, press , choose TESTS, and select 8:TInterval on the calculator. This time, however, choose the Data option. You’ll need to specify which list your data are in, which is L1, and the confidence level (C-Level), which is 0.99 in this example. The value of Freq should be left as the default value of 1. After you select Calculate, you should see the results shown in the screenshot.

  36. Example 8.18: Constructing a Confidence Interval for a Population Mean (Unknown) from Original Data (cont.) Thus, the 99% confidence interval ranges from 0.5240 to 0.7624. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below. We are 99% confident that the mean amount of water used per household for brushing teeth is between 0.5240 and 0.7624 gallons per day.

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