Wheels McGruder, a car insurance representative, believes that the mean age of p

Question

Wheels McGruder, a car insurance representative, believes that the mean age of people who buy their own first car insurance plan is less than 35. He also believes the moon landing was fake and mullets are still cool. To test his car insurance belief he takes a random sample of 15 customers who have just purchased their first car insurance. Their ages are 42, 43, 28, 34, 30, 36, 25, 29, 32, 33, 27, 30, 22, 37, and 40. There is not enough evidence to say the data are nonnormal. Use the provided questions to conclude if Wheels McGruder is right or wrong at a 99% confidence level. What would be the critical value to determine the rejection region?

Explanation / Answer

Consider random variable X: age of people who buy their own first car insurance plan.

Assume X ~ N( mu, sigma^2)

We want to test the hypothesis

H0: mu=35 against H1: mu<35

Since sample size(n)=15 < 30 and population variance is unknown we use one-sample t-test.

Under H0 the test statistic is

t = ( xbar-mu)/(S/sqrt(n) ~ tn-1

where xbar=sum(x)/n and S^2= (1/n-1)*[ sum (x-xbar)^2]

xbar=488/15=32.5333 and S^2= 1/14(533.7333)=38.1238 S= sqrt(38.1238)=6.1744

t=(32.5333-35)/(6.1744/sqrt(15)

= -1.5472

to obtain critical value

alpha = level of significance = 0.01 and d.f=14

by using t-table

critical value = t14,0.01=2.624

Since the alternative hypothesis is left-tailed, we shall apply left-tailed test for testing significance of t.

The critical region for left-tailed test thus consist of all value less than or equal to -2.624

Conclusion: since calculated value (-1.54572) is greater than critical value (-2.624) we accept Ho.

i.e. Wheels McGruder is wrong.

x (x-xbar)^2 42 89.61778 43 109.5511 28 20.55111 34 2.151111 30 6.417778 36 12.01778 25 56.75111 29 12.48444 32 0.284444 33 0.217778 27 30.61778 30 6.417778 22 110.9511 37 19.95111 40 55.75111 488 533.7333

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