Another Probability Question-
A computer manufacturer is about to unveil a new, faster personal computer. The new machine clearly is faster, but initial tests indicate there is more variation in the processing time. The processing time depends on the particular program being run, the amount of input data, and the amount of output. A sample of 16 computer runs, covering a range of production jobs, showed that the standard deviation of the processing time was 22 (hundredths of a second) for the new machine and 12 (hundredths of a second) for the current machine. At the .05 significance level can we conclude that there is more variation in the processing time of the new machine?
Thanks!
To determine if there is more variation in the processing time of the new machine compared to the current machine, we can conduct a hypothesis test using the F-test for comparing variances.
Step 1: Formulate the hypotheses:
- Null hypothesis (H0): There is no more variation in the processing time of the new machine compared to the current machine.
- Alternative hypothesis (Ha): There is more variation in the processing time of the new machine compared to the current machine.
Step 2: Set the significance level:
The significance level, denoted as α, is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Commonly used significance levels are 0.05 or 0.01. In this case, the significance level is given as 0.05.
Step 3: Calculate the test statistic:
The F-test statistic can be calculated by dividing the square of the standard deviation of the new machine (22^2) by the square of the standard deviation of the current machine (12^2).
F = (22^2) / (12^2)
Step 4: Determine the critical value:
The critical value is based on the significance level and the degrees of freedom, which are determined by the sample sizes. Since only the standard deviations are given, you would need to assume equal sample sizes for both machines, which are 16 in this case.
To find the critical value, you can consult an F-distribution table or use statistical software. For example, using a table or software with α = 0.05 and df1 = 15 (sample size - 1) and df2 = 15, the critical value would be approximately 2.772.
Step 5: Make a decision:
Compare the test statistic (F) to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
If F > 2.772, reject the null hypothesis and conclude that there is more variation in the processing time of the new machine.
If F ≤ 2.772, fail to reject the null hypothesis and conclude that there is no significant difference in the variation of the processing time between the new and current machines.
Note: For a more accurate conclusion, it is important to calculate the exact test statistic and compare it to the critical value.