Question

The reasons given by workers for quitting their jobs generally fall into one of two categories:

(1) Worker quits to seek or take a different job, or (2) worker quits to withdraw from the labor force. Economic theory suggests that wages and quit rates are related. The table below lists quit rates (quits per 100 employees) and the average hourly wage in a sample of 15 manufacturing industries.

Industry	Quit Rate,y	Average Wage, x($)
1	1.3	8.45
2	0.6	10.6
3	2.5	6.43
4	3.3	5.62
5	1.6	10.19
6	1.6	9.36
7	0.9	10.84
8	0.4	13.54
9	1.9	8.24
10	3.7	5.79
11	2.2	7.75
12	1.8	6.68
13	1.3	9.08
14	1.7	11.18
15	1.9	9.05

Use Excel to find the regression equation and the coefficient of determination:

1. Do the data present sufficient evidence to conclude that the average hourly wage contributes useful information for the prediction of quit rates? What does your model suggest about the relationship between quit rates and wages?
2. Predict the Quit rate for an average wage of $7.50.
3. Make a scatterplot to represent the data. Comment on the relationship between Quit Rate, y and Average Wage, x.
4. Using excel, find the correlation coefficient r, and coefficient of determination R2 and interpret each.
5. Construct a 95% confidence interval for the population mean quit rate for the industries. Interpret your result.
6. Write a paragraph stating what the results can be used in the industry.

Answer 1

The given data is about the average wage and the quit rate of 15 different industries. The aim is to determine if there is any relation between the two variables.

a) We are to use Excel to determine the regression equation and coefficient of determination. I am using Google Sheets for the purpose. The details remain same with Excel. The data first needs to be enterred in the software. Then using the plot tool, a scatter plot can be constructed. Using the trendline function, we can get the regression line. We can then select on the check boxes to display the equation of the regression line and R2.

Quit Rate,y vs. Average Wage, x($) Quit Rate y -0.347*x + 4.85 R = 0.729 4 3 2 Quit Rate,y . . 0 6 8 10 12 Average Wage, x($)

As can be seen from the trendline, we see that as the average wage of an industry increases, the quit rate decreases. Therefore, the two appear to be inversely proportional.

b) We can predict the quit rate for average wage of $ 7.50, by substituting x = 7.50 in the equation for the regression line. Doing so, we get:

y = -0.347 x 7.50 + 4.85 ..y= -2.6025 + 4.85 Hy 2.25

Therefore, the quit rate for an average wage of 7.50 dollars is 2.25.

c) The scatter plot and the comment on the relationship between the two variables is given above in a.

d) The coefficient of determination is given in the figure above. R2 = 0.729. The correlation coefficient here is r = -0.854.

R2 > 0.05, which implies that the correlation is strong, and r < 0, which states that the correlation between the two variables is negative.

e) Next we are to determine the 95% confidence intervals for the quit rates. For this, we first determine the mean and standard deviation of the quit rate column. The formulae for the same are:

$\\\bar{x} = \frac{\sum_i x_i}{n}\\\\ s = \sqrt{\frac{\sum_i (x_i - \bar{x})^2}{n - 1}}$

Calculating these, we get:

$\\\bar{x} = 1.78\\\\ s = 0.8994$

The number of industries are n = 15.

The formula for the confidence interval is:

$\\CI = \bar{x} \pm z\frac{s}{\sqrt{n}}$

where z is a z-score corresponding to the confidence level. For 95% confidence, z = 1.96.

Substituting the values in the formula, we get:

$\\95\ \%CI = 1.78 \pm 1.96\times \frac{0.8994}{\sqrt{15}}\\\\ \therefore 95\ \%CI = 1.78 \pm 1.96\times \frac{0.8994}{3.8730}\\\\ \therefore 95\ \%CI = 1.78 \pm 1.96\times 0.2322\\\\ \therefore 95\ \%CI = 1.78 \pm 0.4551\\\\ \therefore 95\ \%CI = (1.78 - 0.4551, 1.78 + 0.4551)\\\\ \therefore 95\ \%CI = (1.32, 2.24)$

Therefore, the 95% Confidence interval for the quit rate is between 1.32 and 2.24.

f) The results suggest that there is a significant correlation between the two variables. Also, the correlation is negative which implies that as the average wage increases, the quit rate decreases. Therefore, for industries that are facing large quit rates, they should probably increase the wages of the employees.

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