00:01
For part a here, we are doing a hypothesis test where the null hypothesis is going to be that mu 1 minus mu 2, so the mean value for group a minus the mean value for group b, is less than or equal to, let's see here, less than or equal to the claimed amount, which is 16.
00:25
So we're not testing for the existence of a difference.
00:29
We are testing to see if the difference is greater than or less than 16.
00:36
So our alternate hypothesis is the claim that the yield for variety a exceeds that of variety b by at least 16 bushels per acre.
00:48
So we'd be doing this, well, actually i'll first note for determining the type of test that we need to do.
00:54
We don't have the population standard deviation for either of these, suggesting that we'll want to use a t test.
01:01
When we determine what kind of t test we want to do, we need to test the ratio of the sample standard, or the ratio of the sample variances to determine if we can treat them as coming from a population with the same variance.
01:17
So we have that s1 squared would be 8 .28 to the power of two, and s2 squared would be 7 .62 to the power of two.
01:27
So we get that their ratio is, 1 .18 roughly and well for determining if we can use the pooled variance method we would want to determine if that ratio is between 0 .5 and 2 so we will use the pooled variance or the t test assuming equal variances so to calculate our pooled variance for use in our test that's equal or pardon me our pooled standard deviation that's equal to n1 minus 1 times s1 squared plus n2 minus 1 times s2 squared divided by n1 plus n2 minus 2 so plugging in our values we know both samples were of size 60 so we'd have 59 times 8 .28 squared plus plus 59 times 7 .62 to the power oops not 7 .92 7 .62 to the power of 2, divided by 60 plus 60 minus 2.
02:32
So we get that our pooled standard deviation is equal to about 7 .957 roughly.
02:40
Having that, the value of our test statistic is going to be equal to x bar 1 minus x bar 2.
02:49
Now we need to be careful here.
02:50
It's x bar 1 minus x bar 2 minus the null hypothesized difference divided by our pooled or pooled standard deviation over, or pardon me, not over, but rather times the square root of 1 over n1 plus 1 over n2.
03:09
Plugging in our values, we have that x bar 1 is 86 .7, x bar 2 is 75 .6, and the null hypothesized difference is 16.
03:19
Then we divide that by the square root, or divide that by 7 .957 times the square root of 1 over 60 plus 1 over 60.
03:29
So we get that our test statistic here, our t value, is equal to negative 3 .37 roughly.
03:37
The p value then, for this test statistic, is going to be the probability of a t value with, oh, let me just check here one second.
03:52
Okay, so i just needed to double check here.
03:54
We can automatically see, well, we have a negative t value here, and we have a negative t value here.
04:00
A right -tailed alternate hypothesis.
04:03
So just automatically we should be able to recognize that the data does not support the claim that, or does not support the farmer's claim that the yield for variety a exceeds the yield for variety b by 16 or more.
04:17
That being said, we still want to find our p -value, which is going to be the probability that t is greater than, or pardon me, that is going to be t with n1 plus n2 minus two degrees of freedom so that would be t with 118 degrees of freedom probability of that t value being greater than negative 3 .37 which is going to be a very large probability suggesting very weak evidence for the claim that being said i'm just going to find that value using my software here so student t distribution 118 degrees of freedom um to find the area to the right of negative 3 .37, we can equivalently find the area to the left of positive 3 .37.
05:02
So we can see that, okay, yeah, our p value is 0 .995, which is very strong evidence against the claim...