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Chapter 8

Chapter 8. Technology. Example T.1: Using a TI-84 Plus Calculator to Find the Value of t Given the Area to the Left. Find the value of t for a t -distribution with 17 degrees of freedom such that the area under the curve to the left of t is 0.10 . Solution

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Chapter 8

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  1. Chapter 8 Technology

  2. Example T.1: Using a TI-84 Plus Calculator to Find the Value of t Given the Area to the Left Find the value of t for a t-distribution with 17 degrees of freedom such that the area under the curve to the left of t is 0.10. Solution Press and then to go to the DISTR menu. Choose option 4:invT(. The function syntax for this option is invT(area to the left of t, df). Since the area to the left is the given area in this example, we would enter invT(0.10,17). Press .

  3. Example T.1: Using a TI-84 Plus Calculator to Find the Value of t Given the Area to the Left (cont.) The answer given by the calculator is t −1.333.

  4. Example T.2: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Known) Use a TI-83/84 Plus calculator to find an 85% confidence interval for the population mean amount of money that teachers spend on lunch each week. A survey of 42 randomly selected teachers found that they spend a mean of $18.00 per week on lunch. Assume that the population standard deviation is known to be $2.00 per week.

  5. Example T.2: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Known) (cont.) Solution To begin, press ; then scroll over and choose TESTS. From the TESTS menu, choose option 7:ZInterval. Choose Stats, and then press . Enter 2 for s, 18 for Ë, a sample size of 42 for n, and 0.85 for the level of confidence, C-Level. Highlight Calculate, and then press .

  6. Example T.2: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Known) (cont.) The screen will display the results shown in the screenshot. Thus we are 85% confident that the mean amount of money that teachers spend on lunch each week is between $17.56 and $18.44.

  7. Example T.3: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Unknown) Use a TI-83/84 Plus calculator to find a 95% confidence interval for the population mean amount of money that teachers spend on lunch each week. A survey of 12 randomly selected teachers found that they spend a mean of $20.00 per week on lunch with a sample standard deviation of $3.00 per week.

  8. Example T.3: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) Solution To begin, press ; then scroll over and choose TESTS. From the TESTS menu, choose option 8:TInterval. Choose Stats, and then press . Enter 20 for Ë, 3 for Sx, a sample size of 12 for n, and 0.95 for the level of confidence, C-Level. Highlight Calculate, and then press .

  9. Example T.3: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) The screen will display the results shown in the screenshot. Thus, we are 95% confident that the mean amount of money that teachers spend on lunch each week is between $18.09 and $21.91.

  10. Example T.4: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion A survey of 345 randomly selected students at one university found that 301 students think that there is not enough parking on campus. Find the 90% confidence interval for the proportion of all students at this university who think that there is not enough parking on campus.

  11. Example T.4: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion (cont.) Solution Press ; then scroll over and choose TESTS. From the TESTS menu, choose option A:1-PropZInt. Of those surveyed, 301 students think that there is not enough parking, so enter 301 for x. Our sample size is 345, so enter 345 for n. Our level of confidence is 90%. This must be entered as a decimal, so enter 0.90 for C-level. Choose Calculate and press .

  12. Example T.4: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion (cont.) The screen will display the results shown in the screenshot.

  13. Example T.4: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion (cont.) Notice that the confidence interval is displayed as well as the value of the sample proportion, p̂. Thus, the 90% confidence interval for the proportion of all students at this university who think that there is not enough parking on campus is (0.843, 0.902).

  14. Example T.5: Using Microsoft Excel to Find the Value of tα Find the value of t0.05 for the t-distribution with 15 degrees of freedom. Solution We know the number of degrees of freedom is 15. The area, 0.05, is the area to the right of t, which is the area in one tail of the distribution. Using Microsoft Excel 2010, enter =T.INV(0.05, 15). The value displayed will be approximately -1.753, since the formula assumes that we are entering the area in the left tail. By changing the resulting value of t to a positive number, we obtain t0.05 ≈ 1.753.

  15. Example T.6: Using Microsoft Excel to Find the Value of t Given Area between -t and t Find the critical t-value for a t-distribution with 29 degrees of freedom such that the area between −t and t is 99%. Solution We know the number of degrees of freedom is 29. Since 99% of the area of the curve is in the middle, that leaves 1%, or 0.01 of the area in the two tails. Thus, we need to use the formula for area in two tails, with an area, or probability, of 0.01.

  16. Example T.6: Using Microsoft Excel to Find the Value of t Given Area between -t and t (cont.) Unlike other formulas we have used that required us to divide that area by 2, this formula has that calculation built in. Using Microsoft Excel 2010, enter =T.INV.2T(0.01, 29). The value displayed will be approximately 2.756. This is the positive t-value. Thus, 99% of the area lies between -t ≈ -2.756 and t ≈ 2.756.

  17. Example T.7: Using Microsoft Excel to Find the Margin of Error for a Confidence Interval for a Population Mean (s Known) Calculate the margin of error for a 95% confidence interval for a population mean given that s = 8.5 and n = 47. Solution Since the value for the population standard deviation is given, we want to use a normal distribution to calculate the margin of error. Remember that a = 1 − c, so a = 1 − 0.95 = 0.05. In Excel 2010, enter =CONFIDENCE.NORM(0.05, 8.5, 47). Thus, the margin of error is approximately 2.430066.

  18. Example T.8: Using Microsoft Excel to Find the Margin of Error for a Confidence Interval for a Population Mean (s Unknown) (cont.) Calculate the margin of error for a 95% confidence interval for a population mean given that s = 8.5 and n = 47. Use a t-distribution. Solution Since the value for the population standard deviation is unknown, it is appropriate to use a t-distribution. We are told to use this distribution as well. Remember that a = 1 − c, so a = 1 − 0.95 = 0.05. In Excel 2010, enter =CONFIDENCE.T(0.05, 8.5, 47). Thus, the margin of error is approximately 2.495693.

  19. Example T.9: Using Microsoft Excel to Construct a Confidence Interval for a Population Proportion During the 2010–2011 NBA season, Kobe Bryant attempted 583 free throws and made 483 of these. Construct a 99% confidence interval for his true free throw percentage. Source: FOX Sports on MSN. “NBA Sortable Stats.” 2012. http://msn.foxsports.com/nba/sortableStats?league=NBA&dir=descending&stat=&low=1&high=50&table=freethrows&position=all&showPlayers=min&year=2010&seasonState=regular&Go=Go (22 Jan. 2012).

  20. Example T.9: Using Microsoft Excel to Construct a Confidence Interval for a Population Proportion (cont.) Solution The first step is calculating a point estimate for the interval. Enter the number of successes, 483, into the cell B1 and the total attempts, 583, into B2. We divide the number of successes by the total attempts. Enter =B1/B2 to calculate p̂̂. The margin of error is calculated by entering =NORM.S.INV(0.995)*SQRT(B3*(1-B3)/B2). Finally, subtracting the margin of error from p̂ yields the lower endpoint of the interval and adding the margin of error to p̂ yields the upper endpoint.

  21. Example T.9: Using Microsoft Excel to Construct a Confidence Interval for a Population Proportion (cont.) Thus, based on his performance during the 2010–2011 NBA season, we are 99% confident that Kobe Bryant’s true free throw percentage is between 78.8% and 86.9%.

  22. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) 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 Minitab. Assume that the sample used in the study was a simple random sample.

  23. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.)

  24. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) Solution Since we are not told any population parameters for the study, we cannot assume that s is known or that the population distribution is approximately normal. However, since the sample is large enough, we can use the t-distribution to construct a confidence interval for the population mean. Begin by typing the sample data into the first column, C1.

  25. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) Next, to produce the dialog window for finding a confidence interval for a population mean using a t-distribution, go to Stat ► Basic Statistics ► 1-Sample t. Next, select Samples in columns and enter C1 in the box. Then select Options to enter 99 for the confidence level. Click OK on the options window and OK on the main dialog window and your results will be displayed in the Session window.

  26. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.)

  27. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.)

  28. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) The results in the Session window should appear as follows.

  29. Example T.10: Using Minitab to Find a Confidence Interval for a Population Mean (s Unknown) (cont.) The third column of the results displays the point estimate, x̄, and the last column contains the confidence interval. The 99% confidence interval is (0.5240, 0.7624). Therefore, 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.

  30. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion A certain dosage of insecticide is applied to a sample of 200 cockroaches in a laboratory. The scientists have predicted that this dose should kill more than 50% of the cockroaches. When the results are tallied, 83 cockroaches are dead. Construct a 95% confidence interval for the percentage of cockroaches killed by the insecticide and use the interval to determine whether the evidence supports the scientists’ claim.

  31. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion (cont.) Solution To produce the dialog window for a confidence interval for a population proportion, go to Stat ► Basic Statistics ► 1 Proportion. Next, select Summarized data and enter 83 for the number of events and 200 for the number of trials. Then select Options to enter 95 for the confidence level and select the option, Use test and interval based on normal distribution. Click OK on the options window and OK on the main dialog window and your results will be displayed in the Session window.

  32. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion (cont.)

  33. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion (cont.)

  34. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion (cont.) The results in the Session window should appear as follows.

  35. Example T.11: Using Minitab to Find a Confidence Interval for a Population Proportion (cont.) The fourth column displays the point estimate, p̂, and the fifth contains the confidence interval. The 95% confidence interval, (0.346713, 0.483287), does not contain any values greater than 0.50. Therefore, the scientists’ claim that the tested dosage of insecticide should kill more than 50% of the cockroaches is not supported at a confidence level of 95%.

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