The duration of cigarette smoking has been linked to many...

The duration of cigarette smoking has been linked to many diseases, including lung cancer and various forms of heart disease. Suppose we know that among men ages 30-34 who have ever smoked, the mean number of years they smoked is 12.8 with a standard deviation of 5.1 years. For women in this age group, the mean number of years in 9.3 with standard deviation of 3.2. Assume that the duration of smoking is normally distributed. (4 points) The probability that a man in this age group have smoked for more than 25 years is 0.0793 0.0084 0.9916 0.9207 (4 points) The probability that a woman in this age group have smoked for more than 10 but less than 16 years is 0.1972 0.9819 0.5866 0.3949 (4 points) The 99th percentile of duration of cigarette smoking for men is 2.33 -2.33 24.68 14.32 (4 points) The 15th percentile of duration of cigarette smoking for women is 1.04 -1.04 12.53 5.97

Transcript

00:01 So in this problem, we're given the mean and standard deviation for both men and women for the number of years that they have smoked, if they're smokers.

00:10 So for the men, the mean is 12 .8 years with a standard deviation of 5 .1 years.

00:18 And then for the women, it gives us a mean of 9 .3 years and a standard deviation of 3 .2 years.

00:28 There's a few different things we're asked to find.

00:30 So we'll start here with part a.

00:35 Part a says to assume the duration of smoking is normally distributed, and we're supposed to find the probability that a man in this age group has smoked for more than 25 years.

00:45 So what we're finding here is the probability of x being greater than 25.

00:50 So the first thing we need to do is find a z score for 25 using this red equation i have on the screen.

00:56 So using the equation, z is equal to x, which is 25, minus the mean.

01:02 And make sure that we're looking at the mean for men since we're focusing on men here.

01:06 Then that is 12 .8 divided by the standard deviation, which is 5 .1.

01:13 And this will give us our z score, which comes out to 2 .39.

01:17 So now that we have our z score, we need to turn to our z table on the left side of the screen here and find it.

01:23 So we'll go down on the left -hand column to 2 .3, over on the right to 0 .09.

01:28 And if we meet in the middle of those two points, we see that our probability comes out to 0 .3.

01:35 Now recall our z table always gives us the probability of x being less than the number we're trying to find.

01:41 So this is the probability of x being less than one or less than 25.

01:45 Since we want the probability of x being greater than 25, we'll have to subtract this from one.

01:51 So 1 minus 0 .9916 gives us our answer which is 0 .0084 or about 0 .84 percent.

02:00 So it's a very small probability.

02:04 Now for part b, we're asked to find the probability that a randomly selected woman has smoked for more than 10 years, but less than 16 years.

02:15 So now we're finding the probability of x being between 10 and 16.

02:20 So for this one, we'll have to find z scores for both 10 and 16.

02:25 So let's start with 16.

02:27 We use the same equation we used in part a.

02:29 So z is equal to x, which is 16, minus the mean for women, which is 9 .3.

02:35 Divided by standard deviation for women, which is 3 .2, and that'll give us a z score of 2 .09.

02:43 Now let's do the same for 10.

02:45 This time x is 10...

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