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?)
Take a shot. What do you think. Show some work.
Here's a hint:
Use a confidence interval formula for proportions. Here's one example:
CI99 = p + or - (2.58)(sqrt of pq/n)
...where sqrt = square root, p = x/n, q = 1 - p, and n = sample size.
Remember you need a 90% confidence interval; my example uses a 99% confidence interval. You will need to adjust accordingly.
Another hint: x = 229, n = 525
I hope this will help.
To create a confidence interval for the percentage of females in the labor force in the United States, we can use the sample data provided. Let's begin by calculating the point estimate, which is the proportion of females in the sample.
Point Estimate:
The number of females in the sample is 229, and the total sample size is 525. Therefore, the proportion of females in the sample is 229/525 = 0.4368.
Next, we can calculate the standard error, which measures the variability in our estimate.
Standard Error:
The formula for calculating the standard error of a proportion is:
SE = sqrt((p * (1 - p)) / n)
Where:
- p is the proportion of females in the sample (0.4368)
- n is the sample size (525)
SE = sqrt((0.4368 * (1 - 0.4368)) / 525)
SE ≈ 0.0167
To create the confidence interval, we need to find the margin of error. The margin of error is the maximum likely difference between the point estimate and the true population parameter.
Margin of Error:
We can calculate the margin of error using the following formula:
Margin of Error = Critical Value * Standard Error
The critical value is based on our desired confidence level. Since we want a 90% confidence interval, we need to find the critical value that leaves 5% of the distribution in each tail. This corresponds to a z-score of 1.645.
Margin of Error = 1.645 * 0.0167
Margin of Error ≈ 0.0274
Finally, we can construct the confidence interval by subtracting and adding the margin of error to the point estimate.
Confidence Interval:
Point Estimate ± Margin of Error
0.4368 ± 0.0274
This gives us a confidence interval of (0.4094, 0.4642) or, rounded to two decimal places, approximately (40.94%, 46.42%).
Interpretation:
We are 90% confident that the true percentage of females in the United States labor force is between 40.94% and 46.42%.
Regarding the comparison to European countries, we can see that the confidence interval for the United States does not include the reported 46% from the European countries. However, to definitively conclude whether the percentages are different or not, additional statistical tests would be required.