A study conducted by Hershey's discovered that Americans consumed an average of 11.4 pounds of c

Step 1: For the probability of consuming less than 7 pounds of chocolate next year, we first cal

A study conducted by Hershey's discovered that Americans...

A study conducted by Hershey's discovered that Americans consumed an average of $11.4$ pounds of chocolate per year (seems low to me). Let's assume that the annual chocolate consumption follows the normal distribution with a standard deviation of $3.6$ pounds. a. What is the probability that I will consume 1. less than $7 \mathrm{lb}$. of chocolate next year? 2. more than $9 \mathrm{lb}$. of chocolate next year? 3. between 8 and 12 lb. of chocolate next year? 4. exactly $10 \mathrm{lb}$. of chocolate next year? b. What is the annual consumption of chocolate that represents the 60 th percentile?

Transcript

00:01 All right, so according to the hershey study, the mean amount of chocolate consumed is 11 .4 pounds with a standard deviation of 3 .6.

00:10 So we want the probability of less than 7 pounds of chocolate.

00:16 So we can find the z score associated with it.

00:25 So it's 7 minus 11 .4 over 3 .6, which is negative 4 .4 over 3 .6, which we can then do on a our calculator.

00:43 So negative 1 .222.

00:47 So the area under the curve right here for z equals negative 1 .22 and so on is going to be the same thing as the area above z equals positive 1 .22 by symmetry of the normal curve.

01:04 So if we can find this area, because we only have positive z squared, so i'm doing this, we can find this area.

01:10 We'll take one minus it and that'll give us our answer.

01:12 So for z equals 1 .22, it's 0 .8888.

01:18 So 1 minus 0 .88888 is 0 .112.

01:22 So that's our answer.

01:27 All right.

01:28 Now we have more than 9 pounds of chocolate.

01:36 So again, we can find our z score.

01:40 9 minus 11 .4 over 3 .6 is going to be negative 2 .4 divided by 3 .6.

01:48 So negative 0 .67 .67.

01:52 This time, we want the area above it.

01:57 So if this is z equals negative 0 .67, and this is z equals positive 0 .67, this shaded area is the same thing as the area under the curve for the positive z score.

02:11 So all we have to do is find the area under the curve.

02:13 Look up 0 .67, get the associated percentile.

02:17 So 0 .7486.

02:26 Let me box the answer for part a as well.

02:28 All right, between 8 and 12 pounds of chocolate.

02:43 So for 8, the z score is negative 3 .4 over 3 .6.

02:53 It's going to be negative 0 .94.

02:59 And we know that the area is going to be 1 minus, as we saw from the illustrations, 1 minus the area for the positive z score...

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