Recovery times for patients on an exerciser for people with chronic lower back pain are normally distributed; a random sample of 25 is taken and gave a sample mean of 24.75 and variance of 59.29. Construct an 80% confidence interval for the variance and give the upper limit

Answer 1

To construct an 80% confidence interval for the variance, we will use the Chi-Square distribution.

First, we need to find the critical chi-square values that correspond to the confidence level of 80%. The degrees of freedom (df) for the chi-square distribution in this case is (n - 1), where n is the sample size. Since the sample size is 25, the degrees of freedom is 24.

The lower and upper critical chi-square values can be found using a chi-square calculator or a chi-square table. For an 80% confidence level and 24 degrees of freedom, the lower critical chi-square value is approximately 11.980 and the upper critical chi-square value is approximately 38.079.

Next, we can use these critical chi-square values to calculate the confidence interval for the variance.

The formula for the confidence interval for the variance is:

[ (n - 1) * S^2 / upper chi-square value, (n - 1) * S^2 / lower chi-square value ]

where S^2 is the sample variance.

Plugging in the values, we get:

[ (25 - 1) * 59.29 / 38.079, (25 - 1) * 59.29 / 11.98 ]
[ 24 * 59.29 / 38.079, 24 * 59.29 / 11.98 ]
[ 37.34, 184.47 ]

Therefore, the 80% confidence interval for the variance is [ 37.34, 184.47 ]. The upper limit of the confidence interval is 184.47.