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?
To determine whether 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. Here are the steps to follow:
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that there is no difference in the variation of processing time between the new and current machines.
H0: σ1^2 = σ2^2
The alternative hypothesis (Ha) assumes that there is more variation in the processing time of the new machine.
Ha: σ1^2 > σ2^2
Step 2: Set the significance level.
In this case, the significance level is given as .05.
Step 3: Calculate the test statistic.
The F-test statistic can be calculated using the formula:
F = (s1^2 / s2^2)
where s1^2 is the sample variance of the new machine and s2^2 is the sample variance of the current machine.
Step 4: Determine the critical value.
To determine the critical value, we need the degrees of freedom for each sample. For the new machine, the degrees of freedom (df1) are n1 - 1, and for the current machine, the degrees of freedom (df2) are n2 - 1. In this case, the sample size is 16 for both machines.
Using a significance level of .05 and degrees of freedom (15, 15) for the F-distribution, we can look up the critical value in an F-table or use software.
Step 5: Compare the test statistic to the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Interpret the results.
Based on the comparison in step 5, we can conclude whether there is enough evidence to support the alternative hypothesis or not.
Let's perform the necessary calculations to determine the test statistic and interpret the results.
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.
First, let's define the null and alternative hypotheses:
- Null hypothesis (H0): The variation in processing time of the new machine is the same as the current machine.
- Alternative hypothesis (Ha): The variation in processing time of the new machine is greater than the current machine.
Next, we need to select an appropriate statistical test. Since we are comparing the standard deviations of two samples and looking for a difference in variability, we can use the F-test.
The F-test compares the ratio of the variances of the two samples. The formula to calculate the F-statistic is:
F = (s1^2 / s2^2)
Where s1^2 is the variance of the first sample (new machine) and s2^2 is the variance of the second sample (current machine).
Let's calculate the F-statistic using the given information:
s1 = 22 (standard deviation of the new machine)
s2 = 12 (standard deviation of the current machine)
F = (22^2 / 12^2) = 4.84
Next, we need to determine the critical value for the F-statistic using the significance level of 0.05 and the degrees of freedom.
The degrees of freedom for the numerator (new machine) is (n1 - 1) = (16 - 1) = 15
The degrees of freedom for the denominator (current machine) is (n2 - 1) = (16 - 1) = 15
Using a statistical table or calculator, we find the critical value for an F-distribution with 15 degrees of freedom in the numerator and denominator at a significance level of 0.05 is approximately 2.49.
Finally, we compare the calculated F-statistic (4.84) with the critical value (2.49). If the calculated F-statistic is greater than the critical value, we reject the null hypothesis.
Since 4.84 is greater than 2.49, we reject the null hypothesis. Therefore, we can conclude that there is more variation in the processing time of the new machine compared to the current machine, at the 0.05 significance level.