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For many years, businesses have struggled with the rising cost of health care. But recently, the increases have slowed due to less inflation in health care prices and employees paying for a larger portion of health care benefits. A recent survey showed that 54% ﻿of employers are likely to require higher employee contributions for health care coverage this year relative to last year. Suppose the survey was based on a sample of 600 ﻿companies likely to require higher employee contributions for health care coverage this year relative to last year.

At 95% ﻿confidence, compute the margin of error for the proportion of companies likely to require higher employee contributions for health care coverage. (Round your answer to four decimal places.)

Compute a 95% ﻿confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage. (Round your answers to four decimal places.)


To solve this problem, we need to calculate the margin of error and the 95% confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage.

### Step 1: Calculating the Margin of Error (ME)

The margin of error for a proportion is given by the formula:

\[
\text{Margin of Error (ME)} = Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( \hat{p} \) is the sample proportion
- \( n \) is the sample size
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution for the desired confidence level

For a 95% confidence level, the critical value \( Z_{\alpha/2} \) is approximately 1.96.

Given:
- \( \hat{p} = 0.54 \)
- \( n = 600 \)

### Step 2: Compute the Confidence Interval

The 95% confidence interval is given by:

\[
\text{Confidence Interval} = \hat{p} \pm \text{Margin of Error (ME)}
\]

Let's compute both the margin of error and the confidence interval.### Results:

1. **Margin of Error (ME)**: The margin of error is approximately **0.0399** (rounded to four decimal places).

2. **95% Confidence Interval**: The 95% confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage is **(0.5001, 0.5799)** (rounded to four decimal places).

This means that we are 95% confident that the true proportion of companies that will require higher employee contributions falls between 50.01% and 57.99%.

Do you have any questions or need further details? Here are some related questions you might consider:

1. How do you interpret the margin of error in this context?
2. Why is the confidence interval important in statistical analysis?
3. How would the confidence interval change if the confidence level was increased to 99%?
4. What impact would increasing the sample size have on the margin of error?
5. How does the proportion of \( \hat{p} \) affect the width of the confidence interval?
6. What assumptions are made when calculating the confidence interval for a proportion?
7. Can this method be applied to other surveys with different sample sizes and proportions?
8. How does the confidence interval help businesses make decisions based on the survey data?

**Tip:** Always ensure that the assumptions for calculating confidence intervals (such as randomness of the sample and independence of observations) are met before making inferences based on the results.

Learn how to calculate the margin of error and 95% confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage based on statistical analysis. Includes step-by-step solutions and interpretations. Suitable for undergraduate students and professionals.

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