Question

The director of a state agency believes that the average starting salary for clerical employees in the state is less than ​$32 comma 000 per year. To test her​ hypothesis, she has collected a simple random sample of 100 starting clerical salaries from across the state and found that the sample mean is ​$31 comma 950. a. nbsp State the appropriate null and alternative hypotheses. b. Assuming the population standard deviation is known to be ​$1 comma 500 and the significance level for the test is to be 0.01​, what is the critical value​ (stated in​ dollars)? c. Referring to your answer in part​ b, what conclusion should be reached with respect to the null​ hypothesis? d. Referring to your answer in part​ c, which of the two statistical errors might have been made in this​ case? Explain. a. State the appropriate null and alternative hypotheses. Choose the correct answer below. A. H0​: muequals31 comma 950 HA​: munot equals31 comma 950 B. H0​: mugreater than or equals32 comma 000 HA​: muless than32 comma 000 Your answer is correct.C. H0​: mugreater than or equals31 comma 950 HA​: muless than31 comma 950 D. H0​: muequals32 comma 000 HA​: munot equals32 comma 000 E. H0​: muless than or equals32 comma 000 HA​: mugreater than32 comma 000 F. H0​: muless than or equals31 comma 950 HA​: mugreater than

Answer 1

Solution:

Given:

Claim: the average starting salary for clerical employees in the state is less than ​$32,000 per year.

Thus

Sample size = n = 100

Sample mean =

Population Standard deviation =

Level of significance =

Part a) State the appropriate null and alternative hypotheses.

According to claim H₀ and H_A are:

Vs

Part b) Assuming the population standard deviation is known to be ​$1 comma 500 and the significance level for the test is to be 0.01​, what is the critical value​ (stated in​ dollars)?

It is saying stated in Dollars, so we need to find the sample mean value at which we reject null hypothesis at 0.01 significance level.

Thus first find z value such that: P( Z < z )= 0.01

look in z table for area = 0.0100 or its closest area and find z value.

Area 0.0099 is closest to 0.0100 and it corresponds to -2.3 and 0.03

thus z critical value = -2.33

Now use following formula:

Thus critical value ( in dollars) = $31,650.5

Part c) Referring to your answer in part​ b, what conclusion should be reached with respect to the null​ hypothesis?

Since sample mean = > critical value ( in dollars) = $31,650.5, we do not reject null hypothesis H0. Thus there is not sufficient evidence to conclude that: the average starting salary for clerical employees in the state is less than ​$32,000 per year.

Part d) Referring to your answer in part​ c, which of the two statistical errors might have been made in this​ case? Explain.

There are two types of Errors

Type I Error = Reject H0 , in fact H0 is True.

Type II Error = Fail to reject H0, in fact H0 is False.

Here we failed to reject H0, hence Type II Error might have been made in this​ case.