10-6 Confidence Intervals and Hypothesis Testing

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1 1. LUNCH A sample of 145 high school seniors was asked how many times they go out for lunch per week. The mean number of times was 2.4 with a standard deviation of 0.7. Use a 90% confidence level to calculate the maximum error of estimate stimate z = 1.645, s = 0.7, and n = 145 The maximum error of estimate is approximately Identify the null and alternative hypotheses for each statement. Then identify the statement that represents the claim. 3. Lori thinks it takes a fast-food restaurant less than 2 minutes to serve her meal after she orders it. less than 2 minutes: μ < 2, not less than 2 minutes: μ 2 hypothesis, and make a conclusion about the claim. 7. COMPACT DISCS A manufacturer of blank compact discs claims that each disc can hold at least 84 minutes of music. Using a sample of 219 compact discs, Cayla calculated a mean time of 84.1 minutes per disc with a standard deviation of 1.9 minutes. Test the hypothesis at 5% significance. State the claim and hypothesis. : μ 84 (claim) H a : μ < 84 Determine the critical region. The alternative hypothesis is μ < 84, so this is a lefttailed test. We are testing at 5% significance, so we need to identify the z-value that corresponds with the lower 5% of the distribution. Keystrokes: ` ú InvNorm.05 e The claim is μ < 2, and it is the alternative hypothesis because it does not include equality. The null hypothesis is μ 2, which is the complement of μ < 2. : μ 2 H a : μ < 2 (claim) 5. Mrs. Hart s review game takes at least 20 minutes to complete. at least 20 minutes: μ 20 less than 20 minutes: μ < 20 The claim is μ 20, and it is the null hypothesis because it does include equality. The alternative hypothesis is μ < 20, which is the complement of μ 20. The critical region is z < Calculate the z-statistic for the sample data. z = Formula for z-statistic = = 84.1, µ = 84, s = 1.9, and n = Decide whether to reject the null hypothesis. is not rejected because the z-statistic for the sample does not fall within the critical region. : μ 20 (claim) H a : μ < 20 CCSS RASONING Identify the hypotheses and claim, decide whether to reject the null Make a conclusion about the claim. There is not enough evidence to reject the claim that the compact discs can hold at least 84 minutes of music. esolutions Manual - Powered by Cognero Page 1

2 9. MUSIC A sample of 76 albums had a mean run time of 61.3 minutes with a standard deviation of 5.2 minutes. Use a 95% confidence level to calculate the maximum error of estimate. The sample size n is 76 and the standard deviation s is 5.2. The mean is not needed to calculate the maximum error of estimate. The z-values which correspond to 95% significance are ± A company advertisement states that it takes no more than 2 hours to paint a 200-square-foot room. no more than 2 hours: μ 2 more than 2 hours: μ > 2 The claim is μ 2, and it is the null hypothesis because it does include equality. The alternative hypothesis is μ > 2, which is the complement of μ 2. : μ 2 (claim) H a : μ > 2 The maximum error of estimate is about Identify the null and alternative hypotheses for each statement. Then identify the statement that represents the claim. 11. Julian sends at least six text messages to his best friend every day. at least 6 texts: μ 6 less than 6 texts: μ < 6 The claim is μ 6, and it is the null hypothesis because it does include equality. The alternative hypothesis is μ < 6, which is the complement of μ 6. : μ 6 (claim) H a : μ < 6 esolutions Manual - Powered by Cognero Page 2

3 Identify the hypotheses and claim, decide whether to reject the null hypothesis, and make a conclusion about the claim. 15. PIZZA A pizza chain promises a delivery time of less than 30 minutes. Using a sample of 38 deliveries, Chelsea calculated a mean delivery time of 29.6 minutes with a standard deviation of 3.9 minutes. Test the hypothesis at 1% significance. State the claim and hypothesis. : μ 30 H a : μ < 30 (claim) Determine the critical region. The alternative hypothesis is μ < 30, so this is a lefttailed test. We are testing at 1% significance, so we need to identify the z-value that corresponds with the lower 1% of the distribution. Keystrokes: ` ú InvNorm.01 e 17. DCISION MAKING The number of peaches in 40 random cans is shown below. Should the manufacturer place a label on the can promising exactly 12 peaches in every can? xplain your reasoning. 13, 14, 13, 14, 12, 12, 12, 11, 15, 12, 13, 13, 14, 13, 14, 12, 15, 11, 11, 14, 13, 14, 14, 13, 12, 12, 12, 12, 13, 13, 11, 14, 14, 13, 14, 13, 13, 14, 12, 12 Find the mean and standard deviation of the sample data. nter the data as L1 in a graphing calculator and then select STAT CALC 1 to obtain the statistics. The sample mean and standard deviation are 12.9 and 1.08, respectively. State the claim and hypothesis. : μ = 12 (claim) The critical region is z < Calculate the z-statistic for the sample data. z = Formula for z-statistic = = 29.6, µ = 30, s = 3.9, and n = H a : μ 12 Determine the critical region. The alternative hypothesis is μ 12, so this is a twotailed test. Test at 10% significance. Identify the z- value that corresponds with the upper and lower 5% of the distribution. Keystrokes: ` ú InvNorm.05 e and ` ú InvNorm.95 e. Decide whether to reject the null hypothesis. is not rejected because the z-statistic for the sample does not fall within the critical region. Make a conclusion about the claim. There is not enough evidence to support the claim of a delivery time of less than 30 minutes. The critical region is > z or z > esolutions Manual - Powered by Cognero Page 3

4 Calculate the z-statistic for the sample data. z = Formula for z-statistic = = 12.9, µ = 12, s = 1.08, and n = Decide whether to reject the null hypothesis. is rejected because the z-statistic for the sample falls within the critical region. It would also be rejected at 5% and 1% significance. Make a conclusion about the claim. There is enough evidence to reject the claim that each can contains exactly 12 peaches. Therefore, the company should not put the claim on the label. 19. MULTIPL RPRSNTATIONS In this problem, you will explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where and s =3. a. GRAPHICAL Graph the 90% confidence interval for n = 50, 100, and 250 on a number line. b. ANALYTICAL How does the sample size affect the confidence interval? c. GRAPHICAL Graph the 90%, 95%, and 99% confidence intervals for n = 150. d. ANALYTICAL How does the confidence level affect the confidence interval? e. ANALYTICAL How does decreasing the size of the confidence interval affect the accuracy of the confidence interval? a. n = 50: The 90% confidence interval is µ n = 100: CI = ± = 25 ± 0.70 = 25 and = 0.70 The 90% confidence interval is µ n = 250: 0.49 stimate z = 1.645, s = 3, and n = 100 CI = ± = 25 ± 0.49 = 25 and = stimate z = 1.645, s = 3, and n = 250 CI = ± = 25 ± 0.31 = 25 and = stimate z = 1.645, s = 3, and n = 50 The 90% confidence interval is µ esolutions Manual - Powered by Cognero Page 4

5 b. Sample answer: With everything else held constant, increasing the same size will decrease the size of the confidence interval. c. 90%: CI = ± = 25 ± 0.63 = 25 and = stimate z = 1.645, s = 3, and n = 150 The 99% confidence interval is µ CI = ± = 25 ± 0.40 = 25 and = 0.40 The 90% confidence interval is µ %: d. Sample answer: With everything else held constant, increasing the confidence level will decrease the size of the confidence interval. e. xpanding the confidence interval reduces the accuracy of the estimate. So decreasing the size of the confidence interval increases the accuracy of the estimate stimate z = 1.96, s = 3, and n = 150 CI = ± = 25 ± 0.48 = 25 and = 0.48 The 95% confidence interval is µ %: 0.63 stimate z = 2.576, s = 3, and n = 150 esolutions Manual - Powered by Cognero Page 5

6 21. CHALLNG A 95% confidence interval for the mean weight of a 20-ounce box of cereal was μ with a sample standard deviation of ounces. Determine the sample size that led to this interval. The range of values for the mean weight is to Half of this range is the maximum error of estimate. 25. GOMTRY In the graph below, line l passes through the origin. What is the value of? Use the formula for calculating the maximum error of estimate to determine the same size. The z-values which correspond to 95% significance are ±1.96. A 4 B C D 4 The coordinates of the point O is (0, 0). Find the slope of the line using the points (0, 0) and (1, 4). The slope of the line is. Find the slope of the line using the points (0, 0) and (a, b). 23. WRITING IN MATH How can a statistical test be used in a decision-making process? Sample answer: You can use a statistical test to help you to determine the strength of your decision. But Therefore,. Option C is the correct answer. esolutions Manual - Powered by Cognero Page 6

7 27. The Service Club at Jake s school was founded 8 years ago. The number of members of the club by year is shown in the table. Which linear equation best models the data? 29. HALTH The heights of students at Madison High School are normally distributed with a mean of 66 inches and a standard deviation of 2 inches. Of the 1080 students in the school, how many would you expect to be less than 62 inches tall? Given. The probability that a randomly selected value in the distribution is less than μ 2σ, that is, 66 2(2) or 62. A y = 1.4x B y = 1.4x C y = 1.6x D y = 1.6x Keystrokes: Press STAT NTR 0 NTR 2 NTR 4 NTR 6 NTR 8 NTR 1 1 NTR 1 3 NTR 1 5 NTR 1 9 NTR 2 2 NTR STAT 4 NTR. P(x < 62) = 2% + 0.5% = 2.5% So, 27 students expected to be less than 62 inches tall. Find a n for each geometric sequence. 31. =, r = 3, n = = 16, r = 0.5, n = 8 Option B is the correct answer. Find a 1. Find a 8. esolutions Manual - Powered by Cognero Page 7

8 Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. 35. x 2 = 8y Write an equation in slope-intercept form for each graph. 37. The equation is in standard form of a parabola. Therefore, it represents a parabola. Graph the equation. The line passes trough the origin. Therefore, b = 0. Find the slope of the line using the points (0, 0) and (2.5, 2). The equation of the line is. 39. For all values of x the value of y is 4. Therefore, the equation of the line is y = 4. esolutions Manual - Powered by Cognero Page 8

9 Find each missing measure. Round to the nearest tenth, if necessary. 41. Use the Pythagorean Theorem. a 2 + b 2 = c 2 Pythagorean Theorem = c a = 7 and b = = c 2 = c Take the square root of each side. 25 = c valuate the square root. esolutions Manual - Powered by Cognero Page 9