00:01
For this problem, to begin, i'll note that there's a little bit of missing information from the way that the problem was uploaded, but i was able to find it by doing some searching.
00:10
We have that the sample mean value is equal to 7232, the sample size is equal to 81, and the population standard deviation, sigma, is equal to 1080.
00:29
Now, for part a, we have that the null hypothesis is going to be that the mean value is equal to a stated value from the setup of the problem.
00:42
We're told that the quality control manager at a compact fluorescent light bulb factory needs to determine whether or not the mean life of a large shipment is equal to 7532 hours.
00:55
So that is going to be our null hypothesis.
00:59
And our alternate hypothesis will be that the mean differs from 7532.
01:08
Moving on to finding the test statistic, we have that z -stat is going to be our sample mean minus the hypothesized population mean, divided by sigma over the square root of n.
01:23
Plugging in all of our observed values here, we have 7232 minus 7532, divided by 1080 over the square root of 81, which gives us a result there of negative 2 .5.
01:39
Then, we have that the critical values, since this is a two -tailed test, we have 2, which would be at plus or minus the z -score for a one -tailed proportion of, let's see here, i need to double check what the level of significance is, 0 .05.
02:06
We have plus or minus z, since we're doing two -tailed tests here, with alpha equals 0 .05, our critical values should have a one -tailed proportion of 0 .025.
02:18
And that is going to be, i believe, i want to say it's 1 .96, but i'm just going to double check here.
02:24
So, one -tailed proportion, 0 .025.
02:28
Yes, 1 .96.
02:30
So our critical values are plus or minus 1 .960.
02:34
So, our conclusion then, since we have a magnitude that is greater than positive 1 .96, or alternatively, since we are below negative 1 .96, we will reject the null hypothesis and conclude that there is sufficient evidence to prove that the mean is different from 7532 hours.
02:53
So we would choose option c for our conclusion.
02:56
For part b, to find the p -value, since this is a two -tailed test, it's going to be two times the proportion of the standard normal distribution to the left of our observed value at negative 2 .5...