Suppose a simple random sample of size n = 64 is obtained...

Suppose a simple random sample of size n = 64 is obtained from a population with ? = 72 and ? = 8. (a) Describe the sampling distribution of x?. (b) What is P (x? > 73.65) ? (c) What is P (x? ? 69.9) ? (d) What is P (71.45 < x? < 74.3) ? Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). A. The distribution is skewed left. B. The distribution is approximately normal. C. The distribution is skewed right. D. The distribution is uniform. E. The shape of the distribution is unknown. Find the mean and standard deviation of the sampling distribution of x?. ?x? = ?x? = (b) P (x? > 73.65) = (Round to four decimal places as needed.) (c) P (x? ? 69.9) = (Round to four decimal places as needed.) (d) P (71.45 < x? < 74.3) = (Round to four decimal places as needed.)

Transcript

00:02 So we have a simple random sample of size 64.

00:08 And we're told that the population has a mean of 72 and a standard deviation of 8.

00:15 And we want to find the sampling distribution of the sample mean of x bar.

00:22 And it ends up being that the mean of the sampling distribution is equal to the mean of the population.

00:30 So in this case, it's 72.

00:33 And the standard deviation of the sample about the sample mean, also called sigma sub -x bar, is equal to sigma over root n.

00:49 So this is 8 over the square of 64.

00:53 So it's 8 over 8, which is equal to 1.

00:57 And we can do this piece because the sampling, the sample size is greater than 30.

01:07 And by the central limit theorem, if we have n greater than equal to 30, then the distribution of the sample about the sample mean is normal.

01:19 So that's very handy for us because then we can convert the values we're looking for into z scores.

01:26 So let's go and do that.

01:29 So the first exercise x asks us to find the probability that the mean is greater than 73 .65.

01:41 So what we do is we convert this mean score into a z score.

01:48 And the way we do that is with this formula, x bar equals x bar, or z sub x bar equals x bar minus the mean of the sample distribution, divided by that standard error that we talked about.

02:03 So in terms of how we actually calculate, this is x bar minus 72 divided by less convenient.

02:14 It's just one.

02:15 So it's just x bar minus 72.

02:17 That's our z score.

02:20 So let's go and do that.

02:24 Now the way you figure out, let's do a picture of this actually before we go any further.

02:27 So here's the mean, which is 72.

02:34 We're going to find the probability of being greater than 73 .6 .5.

02:39 So what that means is we want to find this area under our normal distribution.

02:43 And the way we get that is we take 1 minus the probability that the mean is less than 73 .65.

02:55 So we convert that 73 .65 to a z score.

02:59 We end up getting 1 .65.

03:01 We do a little table lookup.

03:02 Or you can use your spreadsheet like i did.

03:05 And i use this function put up here called norm s dist.

03:13 And what you do is you put in your z score and then out will pop the left area but we don't want the left area we're at the right and so we do one minus that left area and that gives us the right and so it's one minus zero point nine five so i'm going to have a four decimals so 0995 which gives us um zero point zero four nine five there we go and for c, we want to find the probability that x bar is less than equal to 69 .9 .9.

04:08 And let's do that little picture of this...

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