Question

Suppose the heights (in inches) of adult males in the United States are normally distributed with a mean of 72 inches and a standard deviation of 2 inches.   Find the percent of men who are no more than 68 inches tall.   The percent of men who are 68 inches tall or below is

Answer 1

The percent of adult males in the United States who are no more than 68 inches tall is approximately 2.28%, as calculated by the z-score and the standard normal distribution table.

Step-by-step explanation:

To find the percent of men who are no more than 68 inches tall, we use the properties of the normal distribution. Given that adult male heights are normally distributed with a mean of 72 inches and a standard deviation of 2 inches, we first calculate the z-score for a height of 68 inches.

The z-score formula is: z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation.

For a height of 68 inches, the z-score is:

z = (68 - 72) / 2
z = -4 / 2
z = -2

This means that a height of 68 inches is 2 standard deviations below the mean. We then consult a standard normal distribution table to find the area to the left of a z-score of -2, which gives us the percent of men who are no more than 68 inches tall. This value is approximately 2.28%, meaning that about 2.28% of adult males in the United States are 68 inches tall or below.

Answer 2

Note that 72-68 = 4, and that 4 is 2 std. dev. from the mean.

68 would be 2 std. dev. below the mean. Alternatively, we could say that the z-score corresponding to a raw score of 68 is -2.

Using either a calculator with built-in stat. functions, or a table of z-scores, we can find the area under the curve to the left of 68. That area is 0.023.

We could also use the Empirical Rule here. That Rule states that 95% of data whose distribution is normal lie within 2 std. dev. of the mean.

This means that (95%/2), or 47.5% of the data lies between 68 and the mean. To the left of the mean is 0.5 of the total area under the curve. So the percentage of males whose heights are no more than 68 inches is 50% less 45.5%, or 2.5% (answer, just slightly different from the previous result).