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 lower limit
To construct an 80% confidence interval for the variance of recovery times, we can use the chi-square distribution.
The chi-square distribution is used for analyzing variances, and its confidence interval formula for the variance is as follows:
[(n - 1) * s^2] / χ^2(a/2, n-1) < σ^2 < [(n - 1) * s^2] / χ^2(1 - a/2, n-1)
Where:
- n is the sample size (25 in this case)
- s^2 is the sample variance (59.29 in this case)
- χ^2(a/2, n-1) is the chi-square critical value at α/2 level of significance with (n-1) degrees of freedom
- χ^2(1 - a/2, n-1) is the chi-square critical value at (1 - α/2) level of significance with (n-1) degrees of freedom
- α is the significance level (1 - confidence level)
Since we need an 80% confidence interval, the significance level (α) is 0.20. Therefore, the confidence level is 1 - 0.20 = 0.80.
Now, we need to find the lower limit of the confidence interval, so we will focus on the left inequality:
[(n - 1) * s^2] / χ^2(a/2, n-1) < σ^2
Substituting the given values:
[(25 - 1) * 59.29] / χ^2(0.20/2, 25-1) < σ^2
Calculating the chi-square critical value:
χ^2(0.20/2, 25-1) = χ^2(0.10, 24) = 35.17
Now we can substitute this value back into the equation:
[(25 - 1) * 59.29] / 35.17 < σ^2
Simplifying the equation:
24 * 59.29 / 35.17 < σ^2
= 40.30 < σ^2
Therefore, the lower limit of the 80% confidence interval for the variance is 40.30 (unit^2).
To construct a confidence interval for the variance of the recovery times, we need to use the chi-square distribution.
Step 1: Determine the degrees of freedom.
The degrees of freedom for estimating the variance is equal to the sample size minus 1.
In this case, the sample size is 25, so the degrees of freedom is 25 - 1 = 24.
Step 2: Find the critical values from the chi-square distribution table.
Since we want an 80% confidence interval, we have a 10% significance level on each tail.
Looking up the critical values in the chi-square distribution table for α = 0.10 and 24 degrees of freedom, we find the critical values as follows:
Lower critical value: χ²(α/2, ν) = χ²(0.10/2, 24) = 12.401
Upper critical value: χ²(1 - α/2, ν) = χ²(1 - 0.10/2, 24) = 41.638
Step 3: Compute the confidence interval for the variance.
The formula for the confidence interval for the variance is:
[(n - 1) * S² / χ²(1 - α/2, ν), (n - 1) * S² / χ²(α/2, ν)]
where n is the sample size, S² is the sample variance, χ²(1 - α/2, ν) is the upper critical value from the chi-square distribution, and χ²(α/2, ν) is the lower critical value.
Plugging in the values from the problem, we get:
Lower limit = (24 * 59.29) / 41.638 = 34.23 (rounded to two decimal places)
Therefore, the lower limit of the 80% confidence interval for the variance is approximately 34.23.