Engineers must consider the breadths of male heads when...

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 5.9-in and a standard deviation of 1.2-in. In what range would you expect to find the middle 50% of most head breadths? Between and . If you were to draw samples of size 48 from this population, in what range would you expect to find the middle 50% of most averages for the breadths of male heads in the sample? Between and . Enter your answers as numbers. Your answers should be accurate to 2 decimal places.

Transcript

00:01 Hi, i'm david and i'm here to help you answer your question.

00:04 Now let me pick up your question here.

00:08 In this question we're going to discuss about the normal distribution.

00:12 Let me remind you that if you have the x followed by the normal with the mean and standard deviation and if we think the x will minus mean over standard deviation, you will follow by the standard normal.

00:25 Also if we discuss about the sum of mean x bar minus mean over standard deviation, but we need to divide by square of n in order to obtain the standard normal.

00:37 Now in this question here we're given x, it will be the potential breath.

00:49 So that will be the head breath and it will follow by the normal when the mean equal to the 5 .9, standard division equal to the 1 .2.

01:07 Now for the question a, want to find the probability that x between the two value a and b, we don't know, such that this probability equal to the 50%, which is 0 .5.

01:23 Now to find this probability, first of all, we have to turn the x into the z.

01:28 In it you do that, we apply the formula here.

01:32 So we have to think the a minus the mean 5 .9.

01:37 By the standard deviation.

01:40 We do the same thing on the right, p minus 5 .9 or 1 .2, equal to 0 .5.

01:48 Now for the 0 .5, we try to draw the graph on the normal distribution, standard normal, and then this will be the 0.

02:00 This will be the value of the, this will be the 50%.

02:08 So the one on the right on the left here also equal to the 0 .25, same thing on the right 0 .25.

02:18 We want to find the value here and here.

02:22 To do it and need to bring up the z table.

02:25 So let me copy the z table here.

02:30 Let me put the table down here for you.