Find the 95% confidence interval for the standard deviation of the lengths of pipes if a sample of 26 pipes has a standard deviation of 10 inches
7.8< o <13.8
95% = mean ± 1.96 SEm
SEm = SD/√n
To find the 95% confidence interval for the standard deviation of the lengths of pipes, we can use the Chi-Square distribution.
Step 1: Determine the degrees of freedom (df).
Since we have a sample size of 26 pipes, the degrees of freedom is given by df = n - 1 = 26 - 1 = 25.
Step 2: Determine the critical values.
To find the critical values for a 95% confidence level, we need to find the chi-square values that enclose 95% of the area under the curve. For a two-tailed test, we split 5% evenly between the upper and lower tails. The critical chi-square values can be found in a chi-square distribution table or using a statistical software. For df = 25 and a significance level of 0.05, the critical values are approximately 39.366 (lower critical value) and 11.143 (upper critical value).
Step 3: Calculate the confidence interval.
The formula to calculate the confidence interval for the standard deviation is given by:
Lower bound = sqrt((n - 1) * s^2 / upper critical value)
Upper bound = sqrt((n - 1) * s^2 / lower critical value)
Substituting the given values, we have:
Lower bound = sqrt((26 - 1) * 10^2 / 11.143)
Upper bound = sqrt((26 - 1) * 10^2 / 39.366)
Calculating the lower bound:
Lower bound = sqrt(25 * 10^2 / 11.143)
= sqrt(2500 / 11.143)
= sqrt(224.5476)
≈ 14.99
Calculating the upper bound:
Upper bound = sqrt(25 * 10^2 / 39.366)
= sqrt(2500 / 39.366)
= sqrt(63.5451)
≈ 7.98
Step 4: Write the final result.
The 95% confidence interval for the standard deviation of the lengths of pipes is approximately 14.99 inches to 7.98 inches.
To find the 95% confidence interval for the standard deviation of the lengths of pipes, we can use the Chi-Square distribution.
The formula to calculate the confidence interval is:
Lower Limit = ((n - 1) * s^2) / Chi-Square Upper
Upper Limit = ((n - 1) * s^2) / Chi-Square Lower
Where:
n is the sample size (26 in this case)
s is the sample standard deviation (10 inches in this case)
Chi-Square Upper and Chi-Square Lower are the upper and lower critical values from the Chi-Square distribution table for a 95% confidence level with (n - 1) degrees of freedom.
For a 95% confidence level and (n - 1) = 25 degrees of freedom, the Chi-Square Upper and Chi-Square Lower values are 41.923 and 12.401 respectively.
Now we can plug in the values:
Lower Limit = ((26 - 1) * 10^2) / 41.923
Upper Limit = ((26 - 1) * 10^2) / 12.401
Lower Limit = (25 * 100) / 41.923
Upper Limit = (25 * 100) / 12.401
Lower Limit ≈ 59.60
Upper Limit ≈ 201.68
Therefore, the 95% confidence interval for the standard deviation of the lengths of pipes is approximately 59.60 inches to 201.68 inches.