00:01
So for this problem, since we are testing to determine if the older proportion is greater than the younger proportion, our null hypothesis is that the two proportions are the same, p1 is equal to p2, and our alternate hypothesis, h1, is going to be that p1 is greater than p2.
00:25
So that corresponds to option e.
00:30
For part b, to find the test statistic, there are a few things we'll need to calculate first.
00:37
First of all, sample proportion 1, p -hat 1, that's the number of successes for the over age 55 group, that's 68 out of 309, let me fix that there, 68 over 309, so that's 0 .22.
00:52
We also need p -hat 2, which is 15 out of the sample of 294, which is 0 .051.
01:10
And we'll need our pooled proportion, which is the total number of people who dream in black and white, 68 plus 15, out of the total sample size, 309 plus 294.
01:23
So we can see that the pooled proportion is 0 .1376, roughly.
01:31
Our test statistic, it's a z -score, is equal to p -hat 1 minus p -hat 2, divided by the square root of p -bar times 1 minus p -bar, times 1 over n1 plus 1 over n2.
01:53
We'll plug in our values, we have 0 .22 minus 0 .051 over the square, oh, let me fix that there, one moment, over the square root of 0 .1376 times 1 minus 0 .1376, times 1 over, pardon me, 1 over 309 plus 1 over 294, which gives a result of 6 .02, roughly.
02:24
For part c, the p -value is going to be the probability of observing a z -score greater than 6 .02, which i'll note that due to the symmetry of the normal distribution that's the same as the proportion of z less than negative 6 .02.
02:48
Now if i go up to a cumulative distribution table, we can see that the proportion of z less than negative 3 .69 is already almost 0 when rounded to four decimal places.
03:00
Going down to negative 6, or equivalently, considering the amount that would be greater than positive 6, we can safely say that that is roughly 0, especially when rounded to three decimal places.
03:15
So the conclusion for the hypothesis test is that since the p -value is less than our level of significance, alpha equals 0 .01, we reject the null hypothesis, there is, one second here, that is reject as one of the things that we put into a blank, and then we would say that there is sufficient, or there is sufficient, or there is evidence to support the hypothesis.
03:48
So we realize that technically that was all part of part a, so let me correct my notation here...