3 complete the diagram below to determine the percentage of...

3) Complete the diagram below to determine the percentage of each portion of the normal distribution. Use the Empirical Rule to help you determine each portion. IT -3σ -2σ -σ μ +σ +2σ +3σ 68% 95% 99.7% 4) In a random selection of 50 freshman, each student's height (in inches) is measured and recorded. The heights follow an approximate normal distribution. The students have a mean height of 65.4 inches and a standard deviation of 2.8 inches. The grades on the exams are all whole numbers, and the grade pattern follows a normal curve. create a sketch of the distribution using the diagram below. a) What percentage of students would have heights between 62.6 inches and 71.0 inches? b) What percentage of students would have heights less than 73.8 inches? c) d) e) What percentage of students would have heights more than 65.4 inches? What percentage of students would have heights between 57.0 and 59.8 inches? Suppose that 120 different freshman are selected. Based on the distribution, how many students would be expected to have a height between 62.6 inches and 68.2 inches?

Transcript

00:02 Okay, so there's four parts to this question.

00:06 The first part says, given that the heights of 300 students are normally distributed with a mean of 68 inches and a standard deviation of three inches, determine how many students have heights greater than 71 inches.

00:22 So here's what we know.

00:24 Our sample size is 300.

00:26 Our mean is 68 inches.

00:32 Our standard deviation is three inches.

00:38 And we're having to determine how many students, let's see, have heights greater than, or x is greater than 71.

00:52 This is our first one.

00:54 Number one says, where is x greater than 71? and then turn it into a number of students, right? so we start by turning this x value.

01:07 Into a z score.

01:09 Since it's normally distributed, we think of the z table.

01:12 So we set up our z score formula, which our z score formula is x minus mu over sigma.

01:22 Okay, so we take our x value 71 minus our mu 68, and then we divide that by sigma, our standard deviation.

01:37 When we put this in our calculator we get a standard deviation of one.

01:42 I'm sorry, a z score of one.

01:46 Z equals one.

01:48 Then what we want to do is look up that z score in the z table to find an area under the curve.

01:54 Okay.

01:55 When we look it up in the table, we find an area of 0 .8413.

02:01 So 84 % of the data is to the left of this z score.

02:09 But the question is, how many? students have a score greater than 71 greater than 71.

02:18 So since this is an area to the left, those are all the scores that are less than 71.

02:25 So we want to take this value and subtract it from one.

02:30 Figure out all the percentage of values that are greater than.

02:35 And then we do that, we get 0 .158, 7.

02:44 So that's the percentage of scores greater than 71.

02:49 Now if i take my sample size, take my sample size, 300, and i multiply that by the percentage of scores that are greater than 71.

03:03 I'll know approximately how many students have a score greater than 71.

03:11 And i'm going to have to round here, so it ends up being 48 students.

03:18 And we're doing a similar thing with all four of these questions.

03:22 Question number two says, it has all the same data, but then it says determine the number of students that have heights less than or equal to 65 inches.

03:35 So in this case, our x value is less than or equal to 65.

03:39 Okay, so we start in the same manner.

03:42 We need to use the z table to determine what percentage of students have scores less than 65.

03:49 So we turn our x value into a z score, and we get negative one.

04:08 Okay.

04:09 We look up the area in the table...