Please help me figure this stats question out

Question

In a survey of a group of​ men, the heights in the​ 20-29 age group were normally​ distributed, with a mean of

67.3

inches and a standard deviation of

4.0

inches. A study participant is randomly selected. Complete parts​ (a) through​ (d) below.

​(a) Find the probability that a study participant has a height that is less than

65

inches.

The probability that the study participant selected at random is less than

65

inches tall is

0.2810.281.

​(Round to four decimal places as​ needed.)

​(b) Find the probability that a study participant has a height that is between

65

and

70

inches.

The probability that the study participant selected at random is between

65

and

70

inches tall is

Answer 1

I like to use the z-score to calculate these probabilities under a normal curve.

z = (x - μ)/σ where x is the value of interest, σ is the std. deviation and μ is the sample mean.

z = (65 - 67.3)/4 = -0.575. The z table should be available in your statistics textbook or on the web.

On the table -0.57 = 0.2843; -0.58 = 0.2810, so -0.575 has a value midway between, or 0.28265, or 0.2827 if rounded to 4 decimal places. That's the probability that a randomly selected participant is less than 65" in height.

For the 2nd part, need to find z-score for 70". z = (70 - 67.3)/4 = That's 0.675. The z table has a value of 0.75015 (have to interpolate again). So the probability of a height between 65 and 70 inches is 0.7502 - 0.2827 = 0.4675.

Answer 2

This is a common normal distribution statistics question.

For part a) Use normalcdf on your graphing calculator. To access normalcdf press 2nd vars, and it should be the second one from the list. After you click normalcdf plug in the lower and upper bounds. The lower bound should be any small number (I usually just put -1000) and the upper bound is 65 because the questions asked height less than 65 inches. Plug in mean which is 67.3 and standard deviation which is 4.0, and you should get 0.2826, or about 28.3% which confirms your answer.

For part b) use normal cdf again, but this time the lower bound is 65 and upper bound is 70. The mean and standard deviation stay the same at 67.3 and 4.0 respectively because the sample didn't change, and you should get about 0.4675 which is roughly 46.8%.