A random sample of n 25 individuals is selected from a...

A random sample of n = 25 individuals is selected from a population with µ = 20, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 22.2 with SS = 384. What was the critical value? What is the test statistic? Did you retain or reject the null? Compute and interpret Cohen's d. Compute the 95% confidence interval.

Transcript

00:01 So for this problem to begin, we know that our ss, presumably that is sum of squared deviations, is equal to 384.

00:09 So to find our sample standard deviation, that's going to be equal to the square root of ss divided by our sample size minus 1.

00:19 So plugging in our values, we have that that's going to be the square root of 384 divided by 19 for a result of 4 .496.

00:33 Then let's see here for, oh excuse me, no the sample size was 25, so that should be divided by 24.

00:41 So we should get a sample standard deviation equal to exactly 4.

00:45 Now regarding the critical value, i'll call this part a even though it's not specifically listed as part a.

00:52 I'll note that with the amount of information that we have, we know that this is going to have to be a t star or a t statistic because well we don't know the population standard deviation.

01:02 But the catch is that we're not given explicitly what our level of significance is, though considering the fact that there is a later portion of the problem where we're asked to use a 95 % confidence interval, i'm going to work with the assumption that our level of significance here is alpha equals 0 .05.

01:24 And again, i additionally need to note that we're not explicitly told what the what hypothesis is being tested here.

01:34 So particularly based on that 95 % confidence interval, i will work under the assumption that the null hypothesis is that the mean value is still equal to 20, and the alternate hypothesis is that it is different from 20, but without more context i can't say anything conclusive.

01:51 That being said, for this our t star value would be t for n minus 1 degrees of freedom, so 24 degrees of freedom, with a tail proportion of half of alpha.

02:01 So if it is indeed alpha equals 0 .05, then we should be looking for t24, 0 .025.

02:08 Now i have a table of values for this.

02:10 We want one tail, 0 .025, 24 degrees of freedom, so i find that that critical value is 2 .064.

02:18 Then i'll call the question about the test statistic part b.

02:28 The test statistic is going to be equal to our sample mean minus the null hypothesized mean value divided by the sample standard deviation over the square root of the sample size...

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