Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 340 grams and a standard deviation of 40 grams. Use the empirical rule to determine the following.

​(a) About 95​% of organs will be between what​ weights?

(b) What percentage of organs weighs between 220 grams and 460 grams?

(c) What percentage of organs weighs less than 220 grams or more than 460 grams?

(d) What percentage of organs weighs between 220 grams and 380 grams?

Answer 1

The weight of an organ in adult males has a bell shaped distribution with a mean of 340 grams, and a standard deviation of 40 grams.

Now, the empircal rule for the normal distribution states that between one standard deviation of the mean, 68% of the data values lie; between two standard deviations of the mean, 95% of the data values lie; between three standard deviations of the mean, 99.7% of the values lie.

Question a

We have to find 95% of organs will be between which weights.

So, according to the empirical rule, 95% of values lie betwee two standard deviations of the mean.

Now,

mean-2*standard deviation=340-2*40=260

mean+2*standard deviation=340+2*40=420

So, 95% of weights lie between 260 and 420 grams.

Question b

We have to find the percentage of organs that fall between 220 grams and 460 grams.

Now, we note that

220=340-3*40=mean-3*standard deviation

460=340+3*40=mean+3*standard deviation

Now, this means that between 3 standard deviations of the mean, 99.7% of the values lie.

So, the correct answer is 99.7%.

Question c

We have to find the percentage of organs that have weight less than 220 grams or more than 460 grams.

This means that we have to percentage of organs that fall outside three standard deviations of the mean.

So, the percentage is 100-99.7, ie. 0.3%.

So, the correct answer is 0.3%.

Question d

We have to find the percentage of weights that follow between 220 grams and 380 grams.

Now, we note

220=340-3*40=mean-3*standard deviation

380=340+40=mean+standard deviation

So, we have to find the percentage of values that fall between three standard deviations to the left of the mean, and one standard deviation to the right of the mean.

Now, between three and one standard deviations, on either side of the mean, lies 99.7-68, ie. 31.7%.

So, on one side lies 31.7/2, ie. 15.85%.

So, the required percentage is

=99.7-15.85

=83.85%.

So, the answer is 83.85%.