Use the given confidence level and sample data to find a confidence interval for the population standard deviation. Assume that a simple random sample has been selected from a population that has a normal distribution.
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80% confidence; n= 91, x = $66,400, s= $ 19,811
Look in the back of your statistics textbook for a table called something like “area under normal distribution” to find the proportion/probability ± .10, and use its Z score.
± Z = (score-mean)/(SD/√n)
Calculate for the Z score both positive and negative.
To find the confidence interval for the population standard deviation, we will use the chi-square distribution. Here are the steps to calculate the confidence interval:
1. Determine the degrees of freedom (df). For finding the confidence interval for the population standard deviation, the degrees of freedom (df) is calculated as n - 1, where n is the sample size. In this case, n = 91, so df = 91 - 1 = 90.
2. Obtain the critical values from the chi-square distribution table. The critical values will depend on both the desired confidence level and the degrees of freedom. For an 80% confidence level, there will be a 10% area of the chi-square distribution in each tail (since 100% - 80% = 20% total area, and divided by 2 gives 10% in each tail). Looking up the critical values in the chi-square distribution table with 90 degrees of freedom, we find that the left critical value is approximately 73.361 and the right critical value is approximately 110.705.
3. Calculate the lower and upper limits of the confidence interval. The formula to calculate the confidence interval for the population standard deviation is:
Lower Limit = sqrt((n-1) * s^2) / sqrt(right critical value)
Upper Limit = sqrt((n-1) *s^2) / sqrt(left critical value)
Plugging in the given values:
Lower Limit = sqrt((91-1) * (19811)^2) / sqrt(110.705) ≈ $17,536.42
Upper Limit = sqrt((91-1) * (19811)^2) / sqrt(73.361) ≈ $27,645.39
So, the 80% confidence interval for the population standard deviation is approximately $17,536.42 to $27,645.39.