The variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance. Jelly Belly Candy Company is testing two machines that use different technologies to fill three pound bags of jelly beans. The file Bags contains a sample of data on the weights of bags (in pounds) filled by each machine. Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for the two machines. Use a 0.05 level of significance. What is your conclusion? Which machine, if either, provides the greater opportunity for quality improvements? Click on the datafile logo to reference the data. DATA variance machine 1= variance machine 2= F=(to 4 decimals) (to 2 decimals) (to 4 decimals)
To conduct a statistical test for comparing the variances of the bag weights for the two machines, we can use the F-test. The F-test compares the ratio of the variances of two populations.
Given the data, we need to calculate the variances for each machine and then the F-statistic to compare the variances. The formulas are as follows:
Variance = (sum of (x - mean)^2) / (n - 1)
F-statistic = variance of machine 1 / variance of machine 2
Let's calculate the variances and F-statistic using the provided sample data.
To test for a significant difference between the variances in the bag weights for the two machines, we can perform an F-test. The F-test compares the variances of two populations.
Let's denote the variance of machine 1 as \( \sigma_1^2 \) and the variance of machine 2 as \( \sigma_2^2 \).
Hypotheses:
- Null hypothesis (\( H_0 \)): There is no significant difference between the variances of the two machines. \( \sigma_1^2 = \sigma_2^2 \)
- Alternative hypothesis (\( H_1 \)): There is a significant difference between the variances of the two machines. \( \sigma_1^2 \neq \sigma_2^2 \)
We can calculate the F-statistic using the formula:
\( F = \frac{{\text{variance of machine 1}}}{{\text{variance of machine 2}}} \)
Let's denote \( F \) as the calculated F-statistic.
The critical F-value can be looked up from the F-distribution table with degrees of freedom (df1 = n1 - 1) and (df2 = n2 - 1), where n1 is the number of bags filled by machine 1 and n2 is the number of bags filled by machine 2.
If the calculated F-statistic is greater than the critical F-value, we reject the null hypothesis and conclude that there is a significant difference in the variances of the two machines. Otherwise, if the calculated F-statistic is less than or equal to the critical F-value, we fail to reject the null hypothesis.
To provide a more thorough analysis, please provide the data file, Bags, so we can perform the calculations and determine the conclusion.