00:01
So in this question, we're told that a simple random sample of 80 items, so n equals 80, resulted in a sample mean of 70, x bar equals 70, and the population standard deviation is sigma equals 5.
00:17
Now in this situation, x bar is going to be normally distributed with mean, the population mean, and standard deviation, the population standard deviation.
00:31
So the probability that...
00:36
So we want to find a confidence interval for the population mean, and what we can do is we can write that the probability that the population mean is in the interval mu 1 to mu 2 is 95%, 0 .95.
01:01
This is going to be the probability that z is in the interval z1, z1 to z2, where z is going to be the mean minus the sample mean, the sample mean, divided by the standard deviation on the sample mean.
01:25
So this is mu minus the sample mean.
01:29
Now the standard deviation on the sample mean is the population standard deviation divided by the square root of n.
01:39
So this, so now what we can do is we can write that this is 0 .9.
01:47
And we can find the symmetric one.
01:49
Now z is normally distributed with means zero and variance 1, which means that the probability that z is in the interval, the symmetric interval, minus z2 to z2.
02:04
Now since z is symmetric, so this is 0 .95.
02:18
So that means that the probability that z is greater than z2 is 0 .0 .025, because that's going to be equal probability in both tails.
02:30
So the probability that z is less than z2 is 0 .975.
02:35
Now we can go into a z score calculator and put in that probability and find that z2 is 1 .96.
02:48
So z1 is minus 1 .96...