7. Two machines are used to fill plastic bottles with...

7. Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volumes are known to be $sigma_1$= 0.10 and $sigma_2$= 0.15 fluid ounces for the two machines, respectively. Two random samples of $n_1$= 12 bottles from machine 1 and $n_2$= 10 bottles from machine 2 are selected, and the sample mean fill volumes are 30.61 fluid ounces from machine 1 and 30.34 fluid ounces for machine 2. a) Test the hypothesis that both machines do not fill to the same mean volume. Use $alpha$=0.05. b) What is the P-value? Interpret this value. c) Construct a 90% two-sided CI on the mean difference. Interpret this interval. d) What is the $eta$-error of the test if the true difference in mean fill volumes is 1.2 fluid ounces?

Transcript

00:01 Hi guys, this problem.

00:02 We need to find a 90 % confidence interval.

00:05 So we know that the 90 % confidence interval is just x1 par minus x2 par plus or minus z of alpha over 2 times sigma 1 squared over n1 plus sigma 2 squared over n2.

00:26 Okay, so after substituting we get this as new, yeah, between all point, this is 0 .19 plus or minus 1 .645 times the square root of 0 .1, 0 .01 over 12, plus 0 .0 .0225, plus 0 .0225 .5.

00:58 Over 10 okay so we can say that as a confidence interval from 0 .0987 to 0 .2813 okay since the confidence interval on the difference in mean fell volume does not include zero and lies to the right of zero there is evidence to conclude that the mean fell volume of the first machine exceeds the mean fell volume of the second machine.

01:32 And we are 90 % confidence that the mean fell volume of the first machine exceeds that of the second one by a number of between these two values.

01:47 So now let's calculate the 95 % confidence interval.

01:53 Okay.

01:54 So it's, yeah, okay.

01:59 It's o .19 plus or minus 1 .96 times the square root which is 0 .01 over 12 plus 0 .0 .225 over 10.

02:20 Okay.

02:21 So this is from 0 .0812 to 0 .2812 to 0 .28888 to to 0 to 988.

02:32 Okay.

02:34 And as the previous one, we are 95 % confident that the fell, mean fell volume of the first machine exceeds that of the second machine, by between one of these two values.

02:48 Okay.

02:49 So the 95 confidence interval is wider than the 95 confidence interval because a higher level of confidence results in a wider confidence interven.

03:03 Okay.

03:04 So for part c, we need to find the upper confidence pound on the difference, such as mu 1 minus mu 2, less than or equal x1 power, which is 30 .88 minus 30 .68 plus z of alpha over 2.

03:27 No it's 0 .05 times the square root of 0 .01 over 12 plus 0 .025 over 10.

03:45 Okay, so we can get that at least than or equal 0 .2813.

03:52 So we are 95 % confident that the mean fell volume of the first machine exceeds that of the second machine by less than 0 .2813...

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