Question

The upper arm lengths of males in United States are approximately Normal with mean 39.1 cm and standard deviation 2.3 cm, i.e., N(39.1, 2.3). Answer the following questions providing all the details.

1. What is the percent of men with Arm-Length longer than 42 cm?

2. At which length x the proportion of men, with Arm-Length shorter than x, is 10%?

3. At which length y the proportion of men, with Arm-Length longer than y, is 15%?

4. What range of lengths covers the middle 99.7% of this Normal distribution?

Answer 1

Let X be the random variable denoting the upper arm length of a randomly selected male in United States.

Hence by the problem,

Part 1

Probability of a man having arm length longer than 42 cm is,

Hence, about 10.4% men have arm length longer than 42 cm.

Part 2

Let x be the length below which 10% men have arm length shorter than that. Then,

Hence, 10% men have arm length shorter than about 36.2 cm.

Part 3

Let y be the length above which 15% men have arm length longer than that. Then,

Hence, about 15% men have arm length longer than 41.5 cm.

Part 4

The 3 standard deviation ban about the mean cover 99.7% of normal distribution.

So, the range is