Question

The manager of a juice bottling factory is considering installing a new juice bottling machine which she hopes will reduce the amount of variation in the volumes of juice dispensed into 8-fluid-ounce bottles. Random samples of 10 bottles filled by the old machine and 9 bottles filled by the new machine yielded the following volumes of juice (in fluid ounces). Old machine: 8.2, 8.0, 7.9, 7.9, 8.5, 7.9, 8.1,8.1, 8.2, 7.9 New machine: 8.0, 8.1, 8.0, 8.1, 7.9, 8.0, 7.9, 8.0, 8.1 Use a 0.05 significance level to test the claim that the volumes of juice filled by the old machine vary more than the volumes of juice filled by the new machine. (Note: you must find s1 and s2)

Answer 1

The sample size is n = 10.

	old machine	old machine2
	8.2	67.24
	8.0	64
	7.9	62.41
	7.9	62.41
	8.5	72.25
	7.9	62.41
	8.1	65.61
	8.1	65.61
	8.2	67.24
	7.9	62.41
Sum =	80.7	651.59

The sample variance s^2 is

Therefore, the sample standard deviation s is

The sample size is n = 9.

	New machine	New machine2
	8.0	64
	8.1	65.61
	8.0	64
	8.1	65.61
	7.9	62.41
	8.0	64
	7.9	62.41
	8.0	64
	8.1	65.61
Sum =	72.1	577.65

The sample variance s^2 is

Therefore, the sample standard deviation ss is

The provided sample variances are s_1^2 = 0.038 and s_2^2 = 0.006 and the sample sizes are given by n_1 = 10 and n_2 = 9.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the rejection region for this right-tailed test is R={F:F>FU​=3.388}.

(3) Test Statistics

The F-statistic is computed as follows:

(4) Decision about the null hypothesis

Since from the sample information we get that F=6.333>FU​=3.388, it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population variance ​ is greater than the population variance , at the α=0.05 significance level.

which means new machine really have less variablity thanold one.

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