Teacher Salaries. Data on salaries in the public school system are published annually in Ranking of the States and Estimates of School Statistics by the National Education Association. The mean annual salary of (public) classroom teachers is \(55.4 thousand. a random sample of 90 classroom teachers in Ohio yielded the annual salaries, in thousands of dollars, presented on the WeissStats site. Us the technology of your choice to perform the following tasks.

a. Obtain a normal probability plot, boxplot, histogram, and stem and leaf diagram of the data.

b. Based on your results from part (a), can you reasonably apply the one-mean z-test to the data? Explain your reasoning.

c. At the 5% significance level, do the data provide sufficient evidence to conclude that the mean annual salary of classroom teachers in Ohio is greater than the national mean? Assume that the standard deviation of annual salaries for all classroom teachers in Ohio is \)9.2 thousand.

given,

The mean annual salary of (public) classroom teachers is $55.4 thousand. a random sample of 90 classroom teachers in Ohio yielded the annual salaries, in thousands of dollars, presented on the WeissStats site. Us the technology of your choice to perform the following tasks.

MINITAB procedure:
Step 1: Choose Graph > Probability Plot.
Step 2: Choose Single, and then click OK.
Step 3: In Graph variables, enter the column of SALARY.
Step 4: Click OK.

Observation:

From the probability plot of the period, the observations are closer to a straight line. Therefore, the distribution of Salary is approximately normally distributed.

From the boxplot, it is clear that the distribution of salary is approximately normally distributed.

MINITAB procedure:

Step 1: Choose Oraph > Histogram.

Step 2: Choose Simple, and then click OK.

Step 3: In Graph variables, enter the corresponding column of SALARY.

Step 4: Click OK.

From the histogram, It is clear that the space of the distribution is bell-shaped.

MINITAB procedure:

Step 1: Select Graph > Stem and leaf.

Step 2: Select the column of variables in Graph variables.

Step 3: Select OK.

Stem-and-Leaf Display: Salary

From the stem-and-leaf diagram, it is clear that the shape of the distribution is bell-shaped.

Yes, it is reasonable apples the one-mean z-test to the data because the distribution of salary is approximately symmetric. That is, the shape of the distribution is bell-shaped. Moreover, the sample size is larger.

State the null and alternative hypotheses:

Null hypothesis:

$H_0:\mu =\$55.4$

That is, the mean annual salary of the classroom teachers in Ohio is not greater than the national mean.

Alternative hypothesis:

$H_0:\mu >\$55.4$

That is, the mean annual salary of the classroom teachers in Ohio is greater than the national mean.

Here, the significance level is, $\alpha =0.05$.

MIN.TAB procedure:

Slep 1: Choose 5tat > Basic Statistics > 1-Sample Z.

Step 2: in Samples in Column, enter the column of SALARY.

Step 3- In Standard deviation, enter 0.92.

Step 4: In-Perform hypothesis test, enter the least mean as 55.4.

Step 5: Check Options, and enter your Confidence level as 95.

Step 6: Choose not equal in alternative.

Step 7: Click OK in all dialogue boxes.

One-Sample Z: SALARY

From the MINITAB output, the value of the test statistic is 1.34 and the P-value is 0.089.

Rejection rule:

If P≤α, then reject the null hypothesis.

Here, the P-value is 0 which is greater than the level of significance. That is P(=0.089)>α(=0.05).

Therefore, the null hypothesis is rejected at the 5% level.

Thus, it can be concluded that the test results are statistically significant at a 5% level of significance

Interpretation:

The data provide sufficient evidence to conclude that the mean annual salary of classroom teachers in Ohio is greater than the national mean at a 5% level.