Chapter 10: Q. 94 (page 603)

94. 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 $$679$ . For $23$ teenage girls, it was $$ 559$ . 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.

Short Answer

Expert verified

Subscripts: $1 =$ boys, $2 =$ girls
(a) The null hypothesis: $H_{0} : μ_{1} \leq μ_{2}$
(b) The alternate hypothesis: $H_{a} : μ_{1} > μ_{2}$
(c) The difference in the mean auto insurance costs for boys and girls is the random variable.
(d) Normal distribution.
(e) The test statistic: $z = 2.50$
(f) The $p$ -value: $0.0062$
(g) Check student's solution.
(h) (i) Alpha: $0.05$
(ii) Decision: Reject the null hypothesis.
(iii) Reason for Decision: $p$ -value $< α$ .
(iv) As a result, there is sufficient evidence to conclude that the mean cost of auto insurance for teenage boys is greater than that for girls at the $5 %$ significance level.

Step by step solution

Given information

To determine the mean cost for auto insurance for teenage boys is greater than that for teenage girls. Then to conduct a hypothesis test to evaluate the manufacturers claim.

Explanation

Subscripts : $1 =$ boys, $2 =$ girls
(a) The null hypothesis is described as follows:
$H_{0} : μ_{1} \leq μ_{2}$
(b) The alternate hypothesis is described as follows:
$H_{a} : μ_{1} > μ_{2}$
(c) The difference in the mean auto insurance costs for boys and girls is the random variable.
(d) Normal distribution.
(e) To determine the test static as:
To access the stat list editor, click STAT and then 1.

Then ENTER all the values the OUTPUT will be:

Hence, the test statistics is $2.497$ .

Explanation

(f) The $p$ -value from the output is determined as $0.00625$ .
(g) Obtain a clear picture of the situation using the information from the last task.
The horizontal axis should be named and measured, and the region(s) on the graph that correlate to the $p$ -value should be shaded.

(h)

(i) $α = 0.05$
(ii) Decision: reject the null hypothesis.
(iii) Reason for Decision: $p -$ value $< α$ .
(iv) As a result, there is enough proof to conclude that the mean cost of auto insurance for teenage boys is more than that for girls, at the $5 %$ level of significance.