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. what type of test will be used in this problem? select an answer
b. identify the null and alternative hypotheses?
$h_0$: select an answer?
$h_a$: select an answer?
c. is the original claim located in the null or alternative hypothesis? select an answer
d. calculate your test statistic. write the result below, and be sure to round your final answer to two decimal places.

Explanation:

Step1: Determine test type

Since the population standard - deviation is unknown and we have a small sample size ($n = 12$), a one - sample t - test will be used.

Step2: Formulate hypotheses

The null hypothesis $H_0$ is the statement of no effect or no difference. The claim is that the mean time in jail is less than 18 months. So, $H_0:\mu\geq18$ and $H_a:\mu < 18$.

Step3: Identify claim location

The original claim "felons convicted of tax fraud spend less than 18 months in jail, on average" is the alternative hypothesis $H_a$.

Step4: Calculate test statistic

The formula for the one - sample t - test statistic is $t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$, where $\bar{x}=16.6$, $\mu = 18$, $s = 2.1$, and $n = 12$.
\[t=\frac{16.6 - 18}{2.1/\sqrt{12}}=\frac{- 1.4}{2.1/\sqrt{12}}\approx\frac{-1.4}{0.6062}\approx - 2.31\]

Answer:

a. One - sample t - test
b. $H_0:\mu\geq18$, $H_a:\mu < 18$
c. Alternative hypothesis
d. - 2.31