The standard deviation of the diameter of 18 baseballs was 0.29cm. Find the 95% confidence interval of the true standard deviation of the diameters of the baseballs. Do you think the manufacturing process should be checked for inconsistency?
To find the 95% confidence interval of the true standard deviation, we can use the chi-square distribution.
First, we need to calculate the chi-square values for the lower and upper limits of the confidence interval.
The chi-square value for the lower limit can be found by using the chi-square distribution with degrees of freedom (df) equal to n-1, where n is the number of baseballs, which in this case is 18.
Using a chi-square table or calculator, we can find the chi-square value that corresponds to a cumulative probability of 0.025 (2.5% divided by 100) for df = 18-1 = 17. Let's call this value chi_lower.
Similarly, the chi-square value for the upper limit can be found using a cumulative probability of 0.975 (97.5% divided by 100) for df = 17. Let's call this value chi_upper.
Now, we can calculate the confidence interval using the formula:
CI = sqrt((n-1)*(s^2) / chi_upper) to sqrt((n-1)*(s^2) / chi_lower)
Where n is the sample size (18) and s is the sample standard deviation (0.29cm).
Plugging in the values:
CI = sqrt((18-1)*(0.29^2) / chi_upper) to sqrt((18-1)*(0.29^2) / chi_lower)
Now, using the chi-square table or calculator, find chi_upper = 33.92 and chi_lower = 7.56 (rounded to two decimal places).
CI = sqrt((17)*(0.29^2) / 33.92) to sqrt((17)*(0.29^2) / 7.56)
Calculating this, we get:
CI = 0.143 to 0.369 (rounded to three decimal places)
The 95% confidence interval for the true standard deviation of the diameters of the baseballs is 0.143cm to 0.369cm.
As for whether the manufacturing process should be checked for inconsistency, it depends on the specific requirements or specifications for the diameter of the baseballs. If the true standard deviation falls within the acceptable range defined by the confidence interval, the manufacturing process may be considered consistent. However, if the true standard deviation is outside the confidence interval, this suggests that the manufacturing process may be inconsistent. Additional investigation or adjustments to the process could be necessary in that case.