You are given a table of paired data below: Pair Sample 1...

You are given a table of paired data below: Pair Sample 1 Sample 2 Differences 1 14 16.7 2 12.9 16 3 10.9 10.2 4 10.1 12.4 5 16.8 18.7 6 15.9 16.4 7 16.2 12.5 8 11.6 10.9 A. Compute the differences = Sample 1 - Sample 2 by filling out the blanks in the table above. B. Fill out the summary statistics for the differences. n = $\bar{d}$ = Round to 4 decimal places. $s_d$ = Round to 4 decimal places. C. Assume that the assumptions for testing are satisfied. Test the following hypotheses using $\alpha$ = 0.05 level of significance. $H_0$: $\mu_d$ = 0, $H_a$: $\mu_d$ ? 0 Test statistic = Round to 4 decimal places. P-value = Round to 4 decimal places. Because P-value is ? $\alpha$ = 0.05, we ? the null hypothesis.

Transcript

00:01 For this problem, in part a, we're told that we are trying to test to see if the experiment increases the value of the observation, or observations, increases the value of the observations.

00:14 So that would mean that we would expect that the after value be greater than the before value.

00:24 If the mean difference is being calculated as after minus before, then that would mean that we would expect that the mean difference is greater than zero if the experiment increases the value of the observations.

00:38 The null hypothesis here should be that there is no effect, or no difference.

00:44 So the mean difference under the null hypothesis is, okay, so this is one thing i'll note here, there are sort of varying different standards for how to phrase a null hypothesis.

00:58 Some books insist that it must always be a strict statement of equality, some insist, or some allow a inequality like we have here.

01:06 In this case, we would have that the correct option out of the ones listed would be the null hypothesis is mu d is less than or equal to zero, and the alternate hypothesis is that mu d is greater than zero.

01:18 So we would go with the third listed option.

01:26 Moving on to part b, calculating the value of the test statistic here, since this is a paired two sample t -test, our test statistic is going to be a t value just equal to the sample mean difference minus the null hypothesized mean difference, which in this case is equal to zero, divided by the standard deviation of the differences over the square root of the number of pairs.

01:58 So what we'll need to do is first find our mean value of the differences, which we find of course by adding up all of the individual pair differences, and then we divide by the number of pairs.

02:10 So you can see in my software here, i've uploaded the data before and after create a new little list, call it d, which is equal to after minus before.

02:19 And so each one of the values here is the result of doing one of the after or the first, or pardon me, it's done pair wise doing after minus before.

02:30 So we have first after minus first before, 2 .9 minus 2 .5 gives 0 .4, then 3 .1 minus 1 .8 giving 1 .3, and so on.

02:39 So finding the sum of those, we have that the total, the numerator there is going to be 12 .8.

02:48 We can see that we have a total of, that looks like, eight measurements.

02:52 So we would have that the mean value is 12 .8 over 8 for a result of 1 .6.

02:57 Then for the standard deviation of the differences, that is the square root of the sum over i of d i minus d bar squared divided by sample size minus 1.

03:15 So going back to my software here, i'll do d minus 1 .6.

03:20 So here we have d1 minus d bar, d2 minus d bar, and so on.

03:26 Then i'll square each one of them.

03:31 So shown now, i have d1 minus d bar squared, d2 minus d bar squared, and so on.

03:37 Then i'll add all of those up.

03:39 So we have that the sum under the square root there is 27 .6...

Question

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