As part of an investigation of employee turnover, an industry-wide survey of people in sales management positions gave the number of years of experience in sales relate positions.

The objective is to construct a 95% confidence interval for the difference between the average years of experience of male and female sales managers.

Let:

• Population 1 (µ₁): Male sales managers
• Population 2 (µ₂): Female sales managers

We need to estimate:

$$\mu_1 - \mu_2$$

Given Data

	Males	Females
Sample size (n)	80	70
Sample mean (x̄)	21.7 years	18.5 years
Standard deviation (s)	9.3 years	4.8 years

Therefore:

$$n_1=80,\quad n_2=70$$ $$\bar{x}_1=21.7,\quad \bar{x}_2=18.5$$ $$s_1=9.3,\quad s_2=4.8$$

The difference between sample means is:

$$\bar{x}_1-\bar{x}_2=21.7-18.5$$ $$=3.2$$

Thus, the sample estimate of the difference in mean experience is 3.2 years.

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Step 1: Formula for Confidence Interval

For two independent samples, the confidence interval for the difference of means is:

$$(\bar{x}_1-\bar{x}_2)\pm Z_{\alpha/2} \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$$

Since the confidence level is 95%:

$$Z_{\alpha/2}=1.96$$

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Step 2: Calculate Standard Error

$$SE=\sqrt{\frac{9.3^2}{80}+\frac{4.8^2}{70}}$$ $$=\sqrt{\frac{86.49}{80}+\frac{23.04}{70}}$$ $$=\sqrt{1.0811+0.3291}$$ $$=\sqrt{1.4102}$$ $$SE\approx1.1875$$

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Step 3: Calculate Margin of Error

$$ME=1.96 \times SE$$ $$=1.96 \times 1.1875$$ $$=2.3275$$

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Step 4: Construct the Confidence Interval

$$(\bar{x}_1-\bar{x}_2)\pm ME$$ $$=3.2\pm2.3275$$

Lower limit:

$$3.2-2.3275=0.8725$$

Upper limit:

$$3.2+2.3275=5.5275$$

Therefore, the 95% confidence interval is:

$$\boxed{0.87 < \mu_1-\mu_2 < 5.53}$$

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Interpretation

We are 95% confident that the true difference between the average years of sales-related experience of male and female sales managers lies between 0.87 years and 5.53 years.

Since the entire confidence interval is positive, it suggests that male sales managers, on average, have more years of experience than female sales managers. The estimated difference is approximately 3.2 years, meaning males in these positions have about three more years of sales-related experience on average.

Thus, the final 95% confidence interval for the difference in population means is:

$$\boxed{(0.87,\ 5.53)\text{ years}}$$