The countries of Europe report that 46% of the labor force is female. The United Nations wonders if the percentage of females in the labor force is the same in the United States. Representatives from the United States Department of Labor plan to check a random sample from 10,000 employment records on file to estimate a percentage of females in the United States labor force. The representatives from the Department of Labor selected a random sample of 525 employment records and found that 229 of the people are females. Create and interpret a 90% confidence interval. (Do you think the percentage of the labor force in the United States is the same as European countries?)

Question

The countries of Europe report that 46% of the labor force is female. The United Nations wonders if the percentage of females in the labor force is the same in the United States. Representatives from the United States Department of Labor plan to check a random sample from 10,000 employment records on file to estimate a percentage of females in the United States labor force. The representatives from the Department of Labor selected a random sample of 525 employment records and found that 229 of the people are females. Create and interpret a 90% confidence interval. (Do you think the percentage of the labor force in the United States is the same as European countries?)

Answer 1

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Answer 2

To create a confidence interval, we can use the sample data to estimate the percentage of females in the United States labor force. The formula to calculate a confidence interval is as follows:

Confidence interval = sample proportion ± (critical value * standard error)

First, let's calculate the sample proportion of females in the United States labor force using the given data. In the sample, 229 out of 525 employment records were found to be females. Thus, the sample proportion is:

Sample proportion = 229/525 ≈ 0.4362

Next, we need to determine the critical value, which is based on the desired confidence level. Since the question asks for a 90% confidence interval, we need to find the critical value corresponding to a 95% level of confidence. This is because the area outside the confidence interval is split equally into two tails, with each tail having a 5% chance (totaling to a 10% chance outside the interval). A common critical value for a 90% confidence level is approximately 1.645 for large sample sizes.

Now, we can calculate the standard error, which measures the average amount by which our estimate is likely to differ from the true population parameter. The formula for the standard error is:

Standard error = sqrt((p * (1-p)) / n)

Where:
p = sample proportion
n = sample size

Substituting the given values into the formula:

Standard error = sqrt((0.4362 * (1 - 0.4362)) / 525) ≈ 0.0174

Finally, we can calculate the confidence interval:

Confidence interval = 0.4362 ± (1.645 * 0.0174)

Simplifying:

Confidence interval ≈ 0.4362 ± 0.0286

This gives us the range 0.4076 to 0.4648.

Interpreting the confidence interval:
We can be 90% confident that the true percentage of females in the United States labor force lies between approximately 40.76% and 46.48%. Since the confidence interval includes the reported European labor force percentage of 46%, it suggests that the percentage of females in the United States labor force could be similar to that of European countries.

However, it's important to note that this is just a sample estimate, and the true population parameter could fall outside the confidence interval. To further investigate and draw conclusions about the percentage of females in the United States labor force compared to other countries, additional studies and data analysis would be needed.