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$680. For 23 teenage girls, it was$564. 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'st-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) Multiple Choice
• State the null hypothesis.

a) H₀:?_boys>?_girls

b) H₀:?_boysgirls

c) H₀:?_boys?_girls

d) H₀:?_boys?_girls

• Part (b) Multiple Choice
• State the alternative hypothesis.

a) H_a:?_boys>?_girls

b) H_a:?_boysgirls

c) H_a:?_boys?_girls

d) H_a:?_boys=?_girls

• Part (c)
• In words, state what your random variableX_boysX_girlsrepresents. (The X's have a line over them)

a) X_boysX_girlsrepresents the average difference in the cost of auto insurance for boys and girls.

b) X_boysX_girlsrepresents the average cost of auto insurance.

c) X_boysX_girls represents the difference in the average cost of auto insurance for boys and girls.

d) X_boysX_girls represents the difference in the cost of auto insurance for boys and girls.

• Part (d) The X's have a line over them
• State the distribution to use for the test. (Round your answers to two decimal places.)
• X_boysX_girls~ _________(__________,__________)
• Part (e)
• What is the test statistic? (If using thezdistribution round your answer to two decimal places, and if using thetdistribution round your answer to three decimal places.)
• (t or z)=___________
• Part (f)
• What is thep-value? (Round your answer to four decimal places.)
• P=__________
• Explain what thep-value means for this problem.

a) IfH₀is true, then there is a chance equal to thep-value that the sample average annual cost of insurance for boys is at least $116 more than the sample average cost for girls.

b) IfH₀is true, then there is a chance equal to thep-value that the sample average annual cost of insurance for boys is $116 less than the sample average cost for girls.

c) IfH₀is false, then there is a chance equal to thep-value that the sample average annual cost of insurance for boys is $116 less than the sample average cost for girls.

d) IfH₀is false, then there is a chance equal to thep-value that the sample average annual cost of insurance for boys is at least $116 more 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 thep-value.
• 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.)
• ?=________
• (ii) Decision:

a) reject the null hypothesis

b) do not reject the null hypothesis

• (iii) Reason for decision:

a) Sincep-value >?, we reject the null hypothesis.

b) Sincep-value

c) Sincep-value >?, we do not reject the null hypothesis.

d) Sincep-value

• (iv) Conclusion:

a) There is sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls.

b) 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) Multiple Choice
• Explain how you determined which distribution to use.

a) Thet-distribution will be used because the samples are independent and the population standard deviation is not known.

b) The standard normal distribution will be used because the samples involve the difference in proportions.

c) Thet-distribution will be used because the samples are dependent.

d) The standard normal distribution will be used because the samples are independent and the population standard deviation is known.