Given a normal distribution with ? = 50 and ? = 4, and given...

Given a normal distribution with ? = 50 and ? = 4, 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) = (Type an integer or decimal rounded to four decimal places as needed.) b. What is the probability that X? is between 49 and 50.5? P(49 < X? < 50.5) = (Type an integer or decimal rounded to four decimal places as needed.) c. What is the probability that X? is above 50.8? P(X? > 50.8) = (Type an integer or decimal rounded to four decimal places as needed.) d. There is a 30% chance that X? is above what value? X? = (Type an integer or decimal rounded to two decimal places as needed.)

Transcript

00:01 This problem tells us that given a normal distribution with the mean 50 and the standard deviation 4, and given that we select a sample of size n equals 100, we are to complete four different questions.

00:16 So first, let's jot down what we were given in the problem.

00:19 So we were given the mean, which is 50, the standard deviation, which is 4, and then the size of the sample, which is n equals 100.

00:30 So to start off with a, a asks what the probability is that x bar is less than 49.

00:42 So the first thing we have to do here is we have to recognize that we are dealing with a sample, and the sample size is larger than 30, n is greater than 30 because it's 100.

00:54 Therefore, the central limit theorem will need to be used, which means that we will have to adjust our population standard deviation to the sample standard deviation.

01:03 The mean will stay the same because the population mean always equals the sample mean, but we'll have to change our standard deviation to account for the sample.

01:13 To do this, we take population standard deviation, which we were given, divided by the square root of n, and we get 0 .4.

01:23 So 0 .4 is what we will use as our standard deviation for this entire problem.

01:29 So now let's start with a.

01:32 So what we need to do here is we need to find the z score for 40.

01:35 Using this red z equation i have here.

01:38 So z is equal to x which is 49, minus the mean which is 50, divided by standard deviation, which we found to be 0 .4.

01:47 That gives us a z score of negative 2 .5.

01:50 So now let's look up negative 2 .5 in our z table.

01:54 So we go down to negative 2 .5 on the left, over to 0 .0 on the top, and we see that we get a probability of 0 .0062.

02:07 Now our z table always gives us what is to the left of the z score or what is less than that, and that is what we're asked in part a.

02:17 We're asked for what is less than 49.

02:19 Therefore, this is our final answer for a, and that's all we have to do.

02:25 Moving on now to b.

02:27 B asks, what is the probability that x bar is between 49 and 50 .5? so, in other words, what's the probability? that x is greater or x bar excuse me is greater than 49 and less than 50 .5 so now we'll need to find the z score for both of these two values so to start with 50 .5 z equals x minus the mean divided by standard deviation and we get 1 .25 as our z score now let's go over to our z table go down to 1 .5 or excuse me, it's 1 .25.

03:09 So we go down to 1 .2 over to 0 .05, and we get a probability of 0 .8944.

03:20 So now we do the same thing, find the z score for 49.

03:24 However, we already did this in part a, so we know that the z score for 49 is going to give us negative 2 .5, and it's going to give us a probability of 0 .0062.

03:35 So now as i mentioned earlier, the z table gives us everything that is to the left of the z score or that is less than it.

03:43 So in order to find what is between 49 and 50 .5, we'll need to subtract what is to the left of 49 from what is to the left of 50 .5...

Question

Please give Ace some feedback