Question

A journalist wants to investigate whether dispersion in sentence lengths for similar crimes concerning narcotics felonies has decreased, due to sentencing guidelines and norms or precedents. The journalist obtains data from the most recent year available, on 61 sentences handed out for felony narcotics convictions (where the defendants had similar histories of arrests and prior convictions). These cases are a random sample of such cases, at the national level. The sample standard deviation is 5 months. Assume that sentence lengths are approximately normally distributed.

The journalist wants to test the null hypothesis that the population standard deviation is at least 5.6 months (that the population variance is at least 31.36 squared months), against the alternative hypothesis that it is less, using a significance level of 5%.

Which of the following is the correct statement of the sample statistic and sampling distribution that this hypothesis test will be based on? LETTER.

The rejection region will be stated in terms of the sample variance, s². Which of the following is the form of the rejection region? LETTER.

A. Reject H₀ if s² < cv
B. Reject H₀ if s² > cv
C. Reject H₀ if s² < cv₁ or if s² > cv₂

What table value(s) do you need to calculate the rejection region? Use 4 decimals.

• Table value if there is one table value: Enter NA for “not applicable” if there are two table values.
• Table values if there are two table values (smaller one first): and Enter NA for “not applicable” in both boxes if there is one table value.

Give the value(s) of the critical value(s). Round to 2 decimals.

• Critical value if there is one critical value: Enter NA for “not applicable” if there are two critical values.
• Critical values if there are two critical values (smaller one first): and Enter NA for “not applicable” in both boxes if there is one critical value.

What is the conclusion of the test? LETTER (see below).

A. Fail to reject H₀ (accept H₀)
B. Reject H₀ in favor of H_A

Consider the hypothesis test you just derived, consisting of either cv, or cv₁ and cv₂, and the rejection rule. Which of the following is TRUE?   LETTER.

The journalist decides to report the results of the hypothesis test described above, using a confidence interval. Which of the following confidence intervals should the journalist calculate? LETTER.

Give the value(s) of the endpoints(s) of the confidence interval. Round to 2 decimals.

• Endpoint if there is one endpoint (L or U): Enter NA for “not applicable” if there are two endpoints.
• Endpoints if there are two endpoints (L and U): and Enter NA for “not applicable” in both boxes if there is one endpoint.

Does the confidence interval you calculated lead to the same conclusion in the hypothesis test as the rejection region, and is this necessary? Choose one of the following. LETTER.

A. The conclusions are the same in this problem, but the conclusions could be different from each other in a different problem.
B. The conclusions are the same and mathematically they always will be the same.
C. The conclusions are different in this problem. In some problems the conclusions are the same and in some they are different.

Answer 1

• The rejection region will be stated in terms of the sample variance, s2. Which of the following is the form of the rejection region?

Answer :  A. Reject H0 if s2 < cv (since the alternative hypothesis is "less than" type)

• What table value(s) do you need to calculate the rejection region? Use 4 decimals.

Answer :

Table value : one table value : FORMULA:  

47.8316

• Give the value(s) of the critical value(s). Round to 2 decimals.  

Answer: Critical value : one critical value : which is equal to -

43.188

• What is the conclusion of the test?

Answer: A. Fail to reject H0 (accept H0) (since critical value < table value)

• Which of the following confidence intervals should the journalist calculate?

Answer: Endpoints :-

      and

So the answer is given below:

L = 18.01

U = 37.05

• Does the confidence interval you calculated lead to the same conclusion in the hypothesis test as the rejection region, and is this necessary?

Answer:- B. The conclusions are the same and mathematically they always will be the same.