Question

a researcher claims that felons convicted of tax fraud spend less than 18 months in jail, on average. she takes a random sample of 12 such cases from court files and finds that the mean is 16.6 months with a standard deviation of 2.1 months. test the claim using a level of significance of 5%. you may assume that the amount of time served by these felons is normally distributed.
a. calculate your p - value. write the result below and be sure to round your final answer to four decimal places.
b. the p - value is

• less than (or equal to)
• greater than

c. do you reject the null hypothesis?

• reject the null hypothesis
• fail to reject the null hypothesis
• accept the null hypothesis

d. what is the correct interpretation of this decision? using a 5% level of significance, there select an answer is not sufficient evidence to select an answer support reject accept the claim that felons convicted of tax fraud spend less than 18 months in jail, on average.

Explanation:

Step1: State the hypotheses

$H_0:\mu\geq18$, $H_1:\mu < 18$.

Step2: Calculate the t - statistic

The formula for the t - statistic is $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$. Here, $\bar{x} = 16.6$, $\mu_0=18$, $s = 2.1$, and $n = 12$.
$t=\frac{16.6 - 18}{2.1/\sqrt{12}}\approx\frac{- 1.4}{2.1/3.4641}\approx\frac{-1.4}{0.6062}\approx - 2.31$.

Step3: Determine the degrees of freedom

The degrees of freedom is $df=n - 1=12-1 = 11$.

Step4: Calculate the p - value

Since this is a left - tailed test, we find the p - value as $P(t_{11}<-2.31)$. Using a t - distribution table or a calculator, the p - value is approximately $0.0228$.

Step5: Compare p - value with significance level

The significance level $\alpha = 0.05$. Since $0.0228<0.05$, the p - value is less than (or equal to) $\alpha$.

Step6: Make a decision

Since the p - value is less than $\alpha$, we reject the null hypothesis.

Step7: Interpret the decision

Using a 5% level of significance, there is sufficient evidence to support the claim that felons convicted of tax fraud spend less than 18 months in jail, on average.

Answer:

a. $0.0228$
b. less than (or equal to)
c. reject the null hypothesis
d. Using a 5% level of significance, there is sufficient evidence to support the claim that felons convicted of tax fraud spend less than 18 months in jail, on average.