A random sample is selected from a normal population with a...

A random sample is selected from a normal population with a mean of μ = 30 and a standard deviation of σ = 8. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 33. A. If the sample consists of n = 16 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = 0.05. B. If the sample consists of n = 64 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = 0.05. C. Comparing your answers for parts A and B, explain how the size of the sample influences the outcome of a hypothesis test.

Transcript

00:01 So we have a random sample from a normal distribution with mean 40 and standard deviation equal to 6.

00:14 After a treatment is administered to the individuals in the sample, we have a sample mean of 37 and we use sample size equal to 36.

00:30 And then we want to see if the treatment is effective or had a significant effect.

00:38 So our known hypothesis would be it has no effect, so the mean hasn't changed.

00:47 The mean is still 40.

00:49 And the alternative would be the mean has changed and that means the treatment had an effect.

00:57 So our sample statistic here then is we have normal, so we can use a z, which is x bar, which in this case is the capital m, minus mu divided by sigma over square root of n.

01:18 So this is 37 minus 40 divided by 6 over the square root of 36.

01:29 So this then is negative 3.

01:33 And we want to use alpha is 0 .05 and do a two -tail test.

01:41 So my critical region is plus or minus z alpha over 2, which is 0 .025, which is 25 % in detail, is positive or negative 1 .960.

02:07 So here we're gonna say our z is greater than z alpha over 2 because, oh, it's less than, we have negative, sorry.

02:26 We are less than negative z alpha over 2, negative 3 and negative 1 .960.

02:35 So if we were to draw this, my rejection region is 1 .960 and negative 1 .960.

02:45 So we're gonna reject anything in detail.

02:48 And our test statistic z is negative 3.

02:55 So that's definitely in the rejection region.

02:58 So conclusion, reject the null hypothesis.

03:06 Treatment has an effect or there is sufficient sufficient evidence to suggest, oops, that treatment has an effect.