A psychologist wants to estimate the variance of employee test scores. A random
sample of 18 scores had a sample standard deviation 10.4. Find a 90% confidence
2
interval for the population variance. What assumption, if any, have you made in calculating this interval estimate?
90% interval = mean ± 1.645 SD
SD^2 = variance
Assume normal distribution.
To find the confidence interval for the population variance, we can use the chi-square distribution. Here are the steps to calculate it:
Step 1: Determine the degrees of freedom (df). For estimating the population variance, the degrees of freedom (df) is n - 1, where n is the sample size. In this case, the sample size is 18, so the degrees of freedom is 18 - 1 = 17.
Step 2: Look up the chi-square critical values for a 90% confidence interval and the given degrees of freedom. From a chi-square table, the critical values for a 90% confidence level and 17 degrees of freedom are approximately 7.564 and 30.191.
Step 3: Calculate the upper and lower limits of the confidence interval.
The lower limit of the confidence interval is given by the formula: (n - 1) * s^2 / chi-square upper critical value. Here, s is the sample standard deviation, which is 10.4, and the chi-square upper critical value is 30.191.
Lower limit = (18 - 1) * 10.4^2 / 30.191
The upper limit of the confidence interval is given by the formula: (n - 1) * s^2 / chi-square lower critical value. Here, the chi-square lower critical value is 7.564.
Upper limit = (18 - 1) * 10.4^2 / 7.564
Step 4: Calculate the values from step 3.
Lower limit = 27.702
Upper limit = 151.576
Step 5: Interpret the confidence interval. The 90% confidence interval for the population variance is (27.702, 151.576). This means that we are 90% confident that the true population variance lies within this interval.
Assumption:
In calculating this interval estimate, we assume that the employee test scores follow a normal distribution within the population.
To find a 90% confidence interval for the population variance, we can use the chi-square distribution. Here's how you can calculate it:
1. First, we need to determine the degrees of freedom (df). Since we are estimating the variance based on a sample, the number of degrees of freedom is equal to the sample size minus 1. In this case, the sample size is 18, so the degrees of freedom (df) is 18 - 1 = 17.
2. Next, we need to find the chi-square critical values for a 90% confidence level. The chi-square distribution is determined by the confidence level and the degrees of freedom. For a 90% confidence level and 17 degrees of freedom, the chi-square critical values are approximately 7.564 and 29.707.
3. Now, we can calculate the confidence interval. The formula for the confidence interval is:
CI = [(n - 1) * s^2 / x2, (n - 1) * s^2 / x1]
where CI is the confidence interval, n is the sample size, s^2 is the sample variance, x1 and x2 are the chi-square critical values.
4. Plug in the values:
CI = [(18 - 1) * (10.4)^2 / 29.707, (18 - 1) * (10.4)^2 / 7.564]
Simplifying this expression will give you the confidence interval for the population variance.
Assumptions made in calculating this interval estimate include:
- The sample is random and representative of the population.
- The population follows a normal distribution.
- The data points in the sample are independent of each other.
It's important to note that these assumptions should be carefully evaluated based on the specific context and data at hand.