Question

A school had decided to hire a new vice principal. A recent issue 4 of the NEA magazine stated that the average vice principal salary s for a school of similar size is $60,000. At the request of the Board of Education, the principal conducts his own survey of vice principal's salaries. His survey of 25 schools revealed an average salary or $72,000 and a standard deviation of $9,000. Perform a hypothesis test at a 5% significance level to test the magazine's claim.

Required:
a. What is the appropriate sampling distribution to use?
b. State the null and alternative hypotheses.
c. State the decision rule at a significance level of 5% show all calculations
d. Illustrate all key numbers and areas on a graph.
e. What decision should the school principal make?
f. What type of error can possibly be made with the decision in part (e)?
g. What is the value of the Maximum Tolerable Error for this problem?

Answer 1

The appropriate sampling distribution to use is the t-distribution. The null hypothesis is that the average vice principal salary for a school of similar size is 60,000. The decision rule at a significance level of 5% is to reject the null hypothesis if the test statistic falls in the critical region.

Step-by-step explanation:

a. What is the appropriate sampling distribution to use?

The appropriate sampling distribution to use in this case is the t-distribution.

b. State the null and alternative hypotheses.

The null hypothesis (H0) is that the average vice principal salary for a school of similar size is 60,000. The alternative hypothesis (Ha) is that the average vice principal salary for a school of similar size is not 60,000.

c. State the decision rule at a significance level of 5% show all calculations.

The decision rule at a significance level of 5% is to reject the null hypothesis if the test statistic falls in the critical region. The critical region is defined by the rejection region when the t-value is less than -2.064 or greater than 2.064.

d. Illustrate all key numbers and areas on a graph.

There are no key numbers or areas to illustrate on a graph for this hypothesis test.

e. What decision should the school principal make?

The school principal should reject the magazine's claim and conclude that the average vice principal salary for a school of similar size is not 60,000.

f. What type of error can possibly be made with the decision in part (e)?

The type of error that can possibly be made with the decision in part (e) is a Type I error. This means that the school principal might conclude that the average vice principal salary for a school of similar size is not 60,000 when it actually is.

g. What is the value of the Maximum Tolerable Error for this problem?

The value of the Maximum Tolerable Error for this problem cannot be determined based on the given information.

Answer 2

1.) t - distribution

2.) H0 : μ = 60000 ; H1 : μ > 60000

3.) P < α (Reject Null) ; P > α (fail to reject Null)

4.) Reject H0

Type 1 error

Step-by-step explanation:

Null hypothesis, H0 : μ = 60000

Alternative hypothesis, H1 : μ > 60000

Decison rule at 5% significance ;

P < α (Reject Null)

P > α (fail to reject Null)

Xbar = 72000 ; std = 9000

Test statistic :

(Xbar - μ) ÷ (std / sqrt(n))

(72000 - 60000) ÷ (9000 / sqrt(25))

12000 ÷ 1800

Test statistic = 6.667

Using the Pvalue calculator from test statistic :

Test statistic = 6.667 ; df = 24 - 1 = 24

Pvalue = 0.00001

Pvalue < α ; Hence, we reject the Null

Type 2 error : This means accepting a Null hypothesis that should be rejected.