The breaking strength of a rivet has mean 9000 psi and...

The breaking strength of a rivet has mean μ = 9,000 psi and standard deviation σ = 450 psi. (a) What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 8,900 psi and 9,200 psi? (b) What is the probability that the sample mean breaking strength for a random sample of 60 rivets is between 8,900 psi and 9,200 psi? (c) Is the probability smaller for n = 40 or n = 60? Briefly explain why this is the case.

Transcript

00:01 This problem tells us that the breaking strength of a rivet has a mean of 9 ,000 psi and a standard deviation of 450 psi.

00:09 So let's just jot that down.

00:12 Mean 9 ,000 standard deviation 450.

00:19 We're asked three different things.

00:22 So let's start off with part a.

00:24 Part a says what is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 8 ,000 ,000.

00:32 900 psi and 9 ,200 psi.

00:36 So what we're finding here is the probability that x is greater than 8 ,900, but less than 9 ,200.

00:48 And here we're also dealing with a sample.

00:51 So up here is our population mean and our population standard deviation, but we need to convert that to a sample mean and a sample standard deviation.

01:00 The sample mean and the sample standard deviation are always the same.

01:03 So we'll still use 9 ,000, nothing to do there, but we will have some things to do to convert the population standard deviation to the sample standard deviation.

01:13 So let's do that first.

01:16 So the sample standard deviation is just the population standard deviation, which we were given, divided by the square root of n, and n is the number in the sample.

01:27 So there's 40 rivets in the sample, so that'll be 40.

01:30 If we calculate that, we get 71 point 15 so 71 .15 is the standard deviation we will use for part a.

01:40 First thing we need to do in part a is to find the z score for both 9 ,200 and 8 ,900.

01:48 So let's start off with 9 ,200.

01:50 So z is equal to x which is 9 ,200 minus the mean which we were given to be 9 ,000 divided by the standard deviation which we found to be 71 .15.

02:04 And that, will give us a z score of 2 .81.

02:09 So now let's go to our z table on the left and find the probability for 2 .81.

02:14 So we go down to 2 .8 on the left and over to 0 .01 on the top.

02:20 And we get probability of 0 .9975.

02:27 Now let's do 8 ,900.

02:31 So that is x minus the mean, and then divided by the standard deviation.

02:39 And that'll give us a z score of negative 1 .41.

02:44 So now let's find that in our table.

02:46 We have to go up to our top table here where the negative z scores are.

02:50 We go down to negative 1 .4 and over to 0 .01, and we can see that we get for our probability 0 .0793.

03:03 Now we are trying to find what is between these two points, the space between them.

03:09 We know that our z table gives us what is to the left of each point...