Find a 95% confidence interval for the true population portion.
In a survey of 319 female employees of a large company, 29% said that they had experienced some form of sexual harassment while working for the company.
To find the 95% confidence interval for the true population proportion, we can use the following formula:
Confidence Interval = Sample Proportion ± (Critical Value) * (Standard Error)
First, let's calculate the sample proportion, which is the proportion of female employees who said they had experienced sexual harassment. In this case, the sample proportion is 29% or 0.29.
The critical value refers to the number of standard deviations away from the mean required to capture a certain percentage of the data. For a 95% confidence level, the critical value is approximately 1.96. This value can be obtained from a standard normal distribution table or calculated using statistical software.
The standard error is a measure of the uncertainty in the sample proportion and is calculated using the following formula:
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
In this case, the sample size is 319.
Let's substitute the values into the formula:
Confidence Interval = 0.29 ± (1.96) * sqrt((0.29 * (1 - 0.29)) / 319)
Now, we can calculate the confidence interval:
Confidence Interval = 0.29 ± (1.96) * sqrt((0.29 * 0.71) / 319)
Confidence Interval = 0.29 ± (1.96) * sqrt(0.2059 / 319)
Confidence Interval = 0.29 ± (1.96) * 0.0202
Confidence Interval = 0.29 ± 0.0396
The 95% confidence interval for the true population proportion is (0.2504, 0.3296).