To evaluate the effect of a treatment a sample of n 8 is...

To evaluate the effect of a treatment, a sample of n = 8 is obtained from a population with a mean of μ = 40, and the treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be M = 35. (Use the 4-step procedure to conduct the hypothesis test) If the sample variance is s2 = 32, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with α = .05? If the sample variance is s2 = 72, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with α = .05? Comparing your answer for parts a and b, how does the variability of the scores in the sample influence the outcome of a hypothesis test?

Transcript

00:01 So this question we're thinking about evaluating the effect of a treatment with a hypothesis test.

00:05 We take a sample of size n equals 8 from a population of mean mu equals 40, and so this is the mean before treatment, and after treatment the sample mean is m equals 35.

00:25 So this is the sample mean after treatment, so this is before in the population, this is after in the sample.

00:37 Now we are looking at whether the treatment has a significant effect.

00:42 So the null is going to be that the mean after is equal to the mean before, which is 40, and the alternate is going to be that the mean after is different from the mean before, so it's not equal to 40.

01:00 So this is two -tailed.

01:04 Now we're using a significance level of 0 .05, and we're doing a student's t -test.

01:12 Sorry, student's t.

01:19 So let's have a look at our rejection region.

01:25 We have a number of degrees of freedom, which is n minus 1, so that's 7.

01:30 So we're going to reject if t is greater than or equal to t star, or t is less than or equal to minus t star, with the probability t is greater than or equal to t star is alpha over 2, so 0 .025, and remember we have seven degrees of freedom.

01:51 This gives t star is equal to, it's the quantile of the t -distribution, enclosing 2 .5 % probability to its right, with seven degrees of freedom, and that's 2 .3646.

02:13 But we also know that t star, sorry, that our t statistic is given by m minus mu divided by s times the square root of n.

02:31 So we can also make this into a rejection region in terms of s.

02:37 S is m minus mu root n divided by t.

02:43 So s star is m minus mu times root n divided by t star.

02:52 So let's see what that is...

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