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?
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To answer this question, we will perform a hypothesis test to compare the variation in processing time for the new machine and the current machine.
The null hypothesis (H0) is that there is no difference in the variation of processing time between the new machine and the current machine. The alternative hypothesis (Ha) is that there is more variation in the processing time of the new machine compared to the current machine.
We will use the F-test to compare the variances of the two samples. The F-statistic is given by:
F = (Standard deviation of new machine)^2 / (Standard deviation of current machine)^2
In this case, we are given that the standard deviation of the new machine is 22 hundredths of a second and the standard deviation of the current machine is 12 hundredths of a second.
Plugging in these values, we can calculate the F-statistic:
F = (22^2) / (12^2) ≈ 4.84
Next, we need to determine the critical value for the F-statistic at a significance level of 0.05. We can look up this value in the F-distribution table or use statistical software.
For a sample size of 16, the degrees of freedom for both the numerator and denominator are 15.
Looking up the critical value in the F-distribution table gives us a value of 2.845.
Since the calculated F-statistic of 4.84 is greater than the critical value of 2.845, we can reject the null hypothesis.
Therefore, we can conclude that there is sufficient evidence to suggest that there is more variation in the processing time of the new machine compared to the current machine at a significance level of 0.05.