   Chapter 7, Problem 7.17P

Chapter
Section
Textbook Problem

P A The city manager has received a complaint from the local union of firefighters to the effect that they are underpaid. Not having much time, the city manager gathers the records of a random sample of 27 firefighters and finds that their average salary is $38 , 073 , with a standard deviation of$ 575 . If she knows that the average salary nationally is $38 , 202 , how can she respond to the complaint? Should she use a one-tailed test in this situation? Why? What would she say in a memo to the union that would respond to the complaint? To determine To find: The significant difference between the given statements. Explanation Given: The given information is, The city manager has received a complaint from the local union of firefighters to the effect that they are underpaid. Not having much time, the city manager gathers the records of a random sample of 27 firefighters and finds that their average salary is$38,073, with a standard deviation of $575. If she knows that the average salary nationally is$38,202, how can she respond to the complaint?

The five step model for hypothesis testing is:

Step 1. Making assumptions and meeting test requirements.

Step 2. Stating the null hypothesis.

Step 3. Selecting the sampling distribution and establishing the critical region.

Step 4. Computing test statistics.

Step 5. Making a decision and interpreting the results of the test.

Formula used:

For small samples with single sample mean and unknown standard deviation, the t value is given by,

t(obtained)=X¯μs/N1

Where, X¯ is the sample mean,

μ is the population mean,

s is the estimate population standard deviation and,

N is the sample size.

Calculation:

From the given information, the sample size is 27, sample mean is $38,073, population mean is$38,202 and the sample standard deviation is \$575.

As the presence of significant difference is the salaries is to be determined, a two tailed test is applied.

Follow the steps for two-tailed hypothesis testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Random sampling.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

The statement of the null hypothesis is that there is no significant difference in the average salary between a sample of firefighters and the population average of firefighters.

Thus, the null and the alternative hypotheses are,

H0:μ=38202

H1:μ38202

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample is small, t distribution can be used.

Thus, the sampling distribution is t distribution.

The level of significance is α=0.05.

The degree of freedom (df) is,

df=N1=271=26

And area of critical region is t(0

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