A study reports that 36 of companies in country a have three...

A study reports that \( 36 \% \) of companies in Country A have three or more female board directors. Suppose you select a random sample of 100 respondents. Complete parts (a) through (c) below. a. What is the probability that the sample will have between \( 34 \% \) and \( 44 \% \) of companies in Country A that have three or more female board directors? The probability is (Round to four decimal places as needed.) b. The probability is \( 80 \% \) that the sample percentage of Country A companies having three or more female board directors will be contained within what symmetrical limits of the population percentage? The probability is \( 80 \% \) that the sample percentage will be contained above \% and below \%. (Round to one decimal place as needed.) c. The probability is \( 95 \% \) that the sample percentage of Country A companies having three or more female board directors will be contained within what symmetrical limits of the population percentage? The probability is \( 95 \% \) that the sample percentage will be contained above \% and below (Round to one decimal place as needed.)

Transcript

00:01 All right, and your question, you're told that 36 % of companies in country a have three or more female board directors.

00:10 So what you're given here is a population proportion, p, of 0 .36.

00:15 We're told that we're going to take a sample of 100, and we're asked to find probabilities about that sample.

00:22 The first thing we would need to do is figure out our standard deviation of our sample proportion.

00:27 And that is found by taking the square root of the population proportion times one minus the population proportion over the sample size.

00:37 So that's going to be the square root of 0 .36 times 1 minus 0 .36 divided by 100.

00:50 And that works out to be 0 .48.

00:54 Then in the first question, part a, i'm going to draw a model out for that.

01:00 We would be interested in our normal distribution of our proportions, our center would be at 0 .36, and we're looking for the probability of being between 0 .34 and 0 .39.

01:15 So this is what you're actually finding, the area under the normal curve.

01:19 To do that, we'll use a program called normal cdf, and it wants your lower boundary, which is 0 .34, then it wants your up.

01:33 Upper boundary, 0 .39, then your mean, which is 0 .36, then your standard deviation, which, sorry, that should have had an extra 0 .08 .08.

01:51 I'm going to type that in, and that works out to be 0 .3956 to four decimal places.

02:13 Now for part b and part c, you're doing a different type of question, so i'm going to draw a different model here to illustrate what we're trying to find.

02:24 Okay, so what we're trying to find here is we know our center is 0 .36.

02:31 We have an x value.

02:33 I'm going to call x sub 1 or a proportion, and we have another proportion call x sub 2.

02:40 And we want to figure out what those values are so that they're symmetrical around the mean, which means the middle, it's in the middle, for this 80 % that we're looking for.

02:53 To do that, we're gonna have to use a different program.

02:58 And the program i'm gonna use to find x1 is gonna be inverse norm.

03:10 And the first thing we have to do is type in the area, which is from the left side of the normal model up to that value.

03:18 So if there's 80 % total in the middle, that means there's 10 % at each tail.

03:24 So for this x1, i'm just gonna be typing in 0 .1 for the area.

03:30 My mean is 0 .36, and my standard deviation is .048.

03:37 For x2, i would be typing in inverse norm, and this time i want the area all the way up to x2.

03:47 That would include that 10 % tail, so we type in .9, 0 .36 .08...