Given a normal distribution with ? = 50 and ? = 8, and given...

Given a normal distribution with ? = 50 and ? = 8, and given you select a sample of n = 100, complete parts (a) through (d). a. What is the probability that X? is less than 49? P(X? < 49) = 0.1056 (Type an integer or decimal rounded to four decimal places as needed.) b. What is the probability that X? is between 49 and 51.5? P(49 < X? < 51.5) = (Type an integer or decimal rounded to four decimal places as needed.) c. What is the probability that X? is above 50.5? P(X? > 50.5) = (Type an integer or decimal rounded to four decimal places as needed.) d. There is a 40% chance that X? is above what value? X? = 50.20 (Type an integer or decimal rounded to two decimal places as needed.)

Transcript

00:01 So for this problem, we are given a normal distribution with mu equals 50 and sigma equals 8.

00:08 We're told that we take a sample of n equals 100.

00:13 So we'd be looking at the sampling distribution, sampling distribution, where the mean value of that sampling distribution is going to be 50.

00:24 And we'd have that the standard deviation of that sampling distribution is going to be equal to the popular.

00:31 Standard deviation divided by the square root of n.

00:35 So that's going to be equal to 8 over 10, or 0 .8 for the standard deviation.

00:43 Once we have all of this information, essentially what we want to do is translate these absolute values of x bar into z scores, so we can compare it against a standard normal table.

00:55 So for part a, we're given an x bar value of 49, so we want z equals 49 minus x bar or minus mu 50, divided by the standard deviation of the sampling mean, or the sample mean, rather.

01:14 So i'll bring up my software for calculation here.

01:16 We have that that is going to give us 49 minus 50 over 0 .8.

01:22 So a z score is negative 1 .25.

01:26 Z equals negative 1 .25.

01:29 And then we can just find the cumulative distribution function for a normal distribution up to that z score of negative 1 .25 to find the probability.

01:40 So we find then that the probability of x bar less than 49 would in turn be equal to the probability of z less than negative 1 .25, which in turn is equal to 0 .1065.

01:57 Five.

01:59 So this procedure of translating the given values into a z score, and then using either a cumulative distribution function or a table or something like that, is essentially the way that you can go through and do these sorts of problems by hand.

02:13 Then in addition, it's possible.

02:16 Once we've made that distribution for the sample means, it's then possible to use technology essentially directly to find the answer for the remaining questions...

Question

Please give Ace some feedback