To conduct a statistical test to determine whether there is a significant difference between the variances in bag weights of Machine I and Machine II, you can use the F-test for comparing variances.
First, let's break down the information in your chart:
- Machine I:
- Mean: 3.3284
- Variance: 0.0489
- Observations: 25
- Degrees of Freedom (df): 24
- Machine II:
- Mean: 3.2782
- Variance: 0.0059
- Observations: 22
- Degrees of Freedom (df): 21
Now, let's proceed with the hypothesis testing:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): There is no significant difference between the variances of bag weights for Machine I and Machine II.
- Alternative hypothesis (Ha): There is a significant difference between the variances of bag weights for Machine I and Machine II.
Step 2: Determine the significance level: In this case, it is given that the significance level (alpha) is 0.05.
Step 3: Calculate the test statistic and critical value:
- The test statistic (F) is given in your chart as 8.2844.
- The critical value of F at the alpha level of 0.05 (one-tail test) with 24 and 21 degrees of freedom is 2.0540.
Step 4: Compare the test statistic and critical value:
- If the test statistic (F) is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In your case, the test statistic (F) is 8.2844, which is greater than the critical value of 2.0540.
Step 5: State the conclusion:
- Since the test statistic is greater than the critical value, we reject the null hypothesis. Therefore, we can conclude that there is a significant difference between the variances of bag weights for Machine I and Machine II.
Step 6: Determine which machine provides the greater opportunity for improvement:
- In this case, since Machine I has a larger variance (0.0489) compared to Machine II (0.0059), Machine I provides a greater opportunity for improvement in reducing the process variance.
In summary, the conclusion is that there is a significant difference between the variances of bag weights for Machine I and Machine II. Machine I provides a greater opportunity for improvement in reducing the process variance.