Question

A business magazine conducted a survey of 751 employees who had been at their current employer for 5 or more years. Of these employees, 295 responded that they were bored in their current position. Create a 99% confidence interval for the proportion of employees who have been with their current employer for 5 or more years and reported being bored in their current position. Use Excel to create the confidence interval, rounding to four decimal places.

Answer 1

Answer:

the $95\%$ confidence interval for the population proportion is:

$\left [0.3469, \hspace{0.1cm} 0.4387\right]$

Step-by-step explanation:

To solve this problem, a confidence interval of $(1-\alpha) \times 100\%$ for the population proportion will be calculated.

$Sample proportion: \bar P=0.3928 \\Sample size n=751 \\Confidence level (1-\alpha)\times100\%=99\% \\ \alpha: \alpha=0.01 \\Z values (for a 99\% confidence) Z_{\alpha/2}=Z_{0.005}=2.5758 \\\\Then, the (1-\alpha) \times 100\% confidence interval for the population proportion is given by:\\\\\left [\bar P - Z_{\frac{\alpha}{2}}\sqrt{\frac{\bar P(1- \bar P)}{n}}, \hspace{0.3cm}\bar P + Z_{\frac{\alpha}{2}}\sqrt\frac{\bar P(1- \bar P)}{n} \right ]$

Thus, the $95\%$ confidence interval for the population proportion is:

$\left [0.3928 - 2.5758\sqrt{\frac{0.3928(1-0.3928)}{751}}, \hspace{0.1cm}0.3928 + 2.5758\sqrt{\frac{0.3928(1-0.3928)}{751}} \right ]=\left [0.3469, \hspace{0.1cm} 0.4387\right]$