Parents of teenage boys often complain that auto insurance costs more, on average, for teenage boys than for teenage girls. A group of concerned parents examines a random sample of insurance bills. The mean annual cost for 36 teenage boys was $677. For 23 teenage girls, it was $556. From past years, it is known that the population standard deviation for each group is $180. Determine whether or not you believe that the mean cost for auto insurance for teenage boys is greater than that for teenage girls. Conduct a hypothesis test at the 5% level. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. O Ho: Hboys girls O Ho: Mboys girls Hoμboys +μgirls O Ho: Mboys >μgirls Part (b) State the alternative hypothesis. Ha Mboys girls Ha Mboys #girls O Ha Mboys girls Ha Hboys girls Part (c) In words, state what your random variable X boys-X girls represents. OX boys-X girls represents the difference in the average cost of auto insurance for boys and girls. OX boys-X girls represents the difference in the cost of auto insurance for boys and girls. OX boys-X girls represents the average cost of auto insurance. OX boys-X girls represents the average difference in the cost of auto insurance for boys and girls. Part (d) State the distribution to use for the test. (Round your answers to two decimal places.) Xboys-Xgirls™ Part (e) What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.) ---Select-- V Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. ○ If Ho is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $121 more than the sample average cost for girls. If Ho is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $121 more than the sample average cost for girls. If Ho is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $121 less than the sample average cost for girls. If Ho is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $121 less than the sample average cost for girls. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. 1/2(p-value) 1/2(p-value) Xboys - X girls Лес p-value Xboys - X girls p-value 1/2(p-value) 1/2(p-value) Xboys - Xgirls Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha (Enter an exact number as an integer, fraction, or decimal.) απ (ii) Decision: O reject the null hypothesis O do not reject the null hypothesis (iii) Reason for decision: O Since p-value a, we reject the null hypothesis. ◇ Since p-value > a, we do not reject the null hypothesis. Xboys - Xgirls (iv) Conclusion: ○ There is sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls. ○ There is not sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls. Part (i) Explain how you determined which distribution to use. The standard normal distribution will be used because the samples involve the difference in proportions. The t-distribution will be used because the samples are dependent. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples are independent and the population standard deviation is known.

Parents of teenage boys often complain that auto insurance costs more, on average, for teenage boys than for teenage girls. A group of concerned parents examines a random sample of insurance bills. The mean annual cost for 36 teenage boys was $677. For 23 teenage girls, it was $556. From past years, it is known that the population standard deviation for each group is $180. Determine whether or not you believe that the mean cost for auto insurance for teenage boys is greater than that for teenage girls. Conduct a hypothesis test at the 5% level. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. O Ho: Hboys girls O Ho: Mboys girls Hoμboys +μgirls O Ho: Mboys >μgirls Part (b) State the alternative hypothesis. Ha Mboys girls Ha Mboys #girls O Ha Mboys girls Ha Hboys girls Part (c) In words, state what your random variable X boys-X girls represents. OX boys-X girls represents the difference in the average cost of auto insurance for boys and girls. OX boys-X girls represents the difference in the cost of auto insurance for boys and girls. OX boys-X girls represents the average cost of auto insurance. OX boys-X girls represents the average difference in the cost of auto insurance for boys and girls. Part (d) State the distribution to use for the test. (Round your answers to two decimal places.) Xboys-Xgirls™ Part (e) What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.) ---Select-- V Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. ○ If Ho is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $121 more than the sample average cost for girls. If Ho is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $121 more than the sample average cost for girls. If Ho is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $121 less than the sample average cost for girls. If Ho is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $121 less than the sample average cost for girls. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. 1/2(p-value) 1/2(p-value) Xboys - X girls Лес p-value Xboys - X girls p-value 1/2(p-value) 1/2(p-value) Xboys - Xgirls Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha (Enter an exact number as an integer, fraction, or decimal.) απ (ii) Decision: O reject the null hypothesis O do not reject the null hypothesis (iii) Reason for decision: O Since p-value a, we reject the null hypothesis. ◇ Since p-value > a, we do not reject the null hypothesis. Xboys - Xgirls (iv) Conclusion: ○ There is sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls. ○ There is not sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls. Part (i) Explain how you determined which distribution to use. The standard normal distribution will be used because the samples involve the difference in proportions. The t-distribution will be used because the samples are dependent. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples are independent and the population standard deviation is known.

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015

1st Edition

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:HOUGHTON MIFFLIN HARCOURT

Chapter11: Data Analysis And Displays

Section: Chapter Questions

Problem 5CT

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