Questions: Employee Pay Forty-three percent of employers said that their employees are fairly paid, while only 20% of the employees agreed. If 50 employers and 200 employees were surveyed, find the 90% confidence level of the proportions for each group. Round your answers to at least three decimal places.

Employee Pay Forty-three percent of employers said that their employees are fairly paid, while only 20% of the employees agreed. If 50 employers and 200 employees were surveyed, find the 90% confidence level of the proportions for each group. Round your answers to at least three decimal places.

Transcript text: Employee Pay Forty-three percent of employers said that their employees are fairly paid, while only $20 \%$ of the employees agreed. If 50 employers and 200 employees were surveyed, find the $90 \%$ confidence level of the proportions for each group. Round your answers to at least three decimal places.

Solution

Solution Steps

Step 1: Given Data

We are given the following data for the proportion of employers who believe their employees are fairly paid:

Proportion of employers ($\hat{p}$): $0.43$
Sample size of employers ($n$): $50$
Confidence level: $90\%$

Step 2: Calculate the Z-Score

For a $90\%$ confidence level, the Z-score ($z$) corresponding to the critical value is approximately $1.645$.

Step 3: Calculate the Standard Error

The standard error (SE) for the proportion is calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.43(1 - 0.43)}{50}} = \sqrt{\frac{0.43 \cdot 0.57}{50}} \approx 0.070 \]

Step 4: Calculate the Margin of Error

The margin of error (ME) is calculated as: \[ ME = z \cdot SE = 1.645 \cdot 0.070 \approx 0.115 \]

Step 5: Calculate the Confidence Interval

The confidence interval is given by: \[ \hat{p} \pm ME = 0.43 \pm 0.115 \] This results in: \[ (0.43 - 0.115, 0.43 + 0.115) = (0.315, 0.545) \]