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
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.
To construct a confidence interval for the variance, we can use the chi-square distribution. The formula for the confidence interval is:
[ (n - 1) * s^2 / chi-square upper, (n - 1) * s^2 / chi-square lower ]
where n is the sample size, s^2 is the sample variance, and chi-square upper and chi-square lower correspond to the upper and lower critical values from the chi-square distribution.
In this case, we have:
n = 25
s^2 = 59.29
To find the critical values from the chi-square distribution, we need to determine the degrees of freedom. The degrees of freedom for estimating the variance is (n - 1), which in this case is (25 - 1) = 24.
Using a chi-square table with 24 degrees of freedom and an 80% confidence level, we find that the upper critical value is approximately 36.415.
Substituting the values into the formula, we get:
[(25 - 1) * 59.29 / 36.415, (25 - 1) * 59.29 / chi-square lower]
Simplifying:
[24 * 59.29 / 36.415, 24 * 59.29 / chi-square lower]
Calculating the values:
[39.11, 63.75]
Therefore, the 80% confidence interval for the variance is [39.11, 63.75], and the upper limit is 63.75.
To construct a confidence interval for the variance, we can use the chi-square distribution.
Here are the steps to construct an 80% confidence interval for the variance:
Step 1: Determine the degrees of freedom (df).
Since we have a sample size of 25, the degrees of freedom is calculated as (n - 1), where n is the sample size. Therefore, in this case, the degrees of freedom is 24.
Step 2: Determine the critical values from the chi-square distribution.
To determine the critical values, we need the chi-square value that corresponds to an alpha level of 0.1 (since the confidence level is 80%). We can find this value using a chi-square distribution table or a statistical calculator. The critical values for an 80% confidence level with 24 degrees of freedom are 9.886 and 37.652.
Step 3: Calculate the confidence interval.
The confidence interval for the variance is calculated using the formula:
[(n - 1) * sample variance / chi-square upper, (n -1) * sample variance / chi-square lower]
where "n" is the sample size, "sample variance" is the variance of the sample, and "chi-square upper" and "chi-square lower" are the critical values from Step 2.
In this case, the sample variance is 59.29.
So, the confidence interval for the variance is:
[(24 - 1) * 59.29 / 37.652, (24 - 1) * 59.29 / 9.886]
[35.006, 214.558]
The upper limit of the 80% confidence interval for the variance is 214.558.