Question

A large study of the heights of 1170 adult men found that the mean height was 71 inches tall. The standard deviation was 8 inches. If the distribution of data was normal, what is the probability that a randomly selected male from the study was between 63 and 87 inches tall? Use the 68-95-99.7 rule (sometimes called the Empirical rule or the Standard Deviation rule). For example, enter 0.68, NOT 68 or 68%.

Round your answer to three decimal places.

Caution: Using tables or Excel for this may produce a wrong answer. Use the 68-95-99.7 rule.

Answer 1

Ans:

Given that

mean=71

standard deviation=8

63 is 1 standard deviation below the mean

87 is 2 standard deviations above the mean

we know that 68% of the data falls within one standard deviation and 95% data falls within 2 standard deviation of the mean.

So,Probability that a randomly selected male from the study was between 63 and 87 inches tall

=(0.68+0.95)/2

=0.815

Answer 2

The 68-95-99.7 rule, also known as the Empirical rule or the Standard Deviation rule, states that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.

• Approximately 95% of the data falls within two standard deviations of the mean.

• Approximately 99.7% of the data falls within three standard deviations of the mean.

In this case, the mean height is 71 inches and the standard deviation is 8 inches.

To calculate the probability that a randomly selected male from the study has a height between 63 and 87 inches, we can use the 68-95-99.7 rule.

First, let's find the range within two standard deviations of the mean: Lower limit = mean - 2 * standard deviation Lower limit = 71 - 2 * 8 Lower limit = 55 inches

Upper limit = mean + 2 * standard deviation Upper limit = 71 + 2 * 8 Upper limit = 87 inches

The range from 55 to 87 inches covers approximately 95% of the data.

Since we want to find the probability that a randomly selected male falls between 63 and 87 inches, we need to find the proportion of the 95% range that falls within this specific range.

To do this, we calculate the z-scores for the lower and upper limits of the range: Z1 = (63 - mean) / standard deviation Z2 = (87 - mean) / standard deviation

Using the formula, we have: Z1 = (63 - 71) / 8 = -1 Z2 = (87 - 71) / 8 = 2

Now, we can consult the standard normal distribution table or use a calculator to find the proportion of data between -1 and 2 standard deviations.

From the standard normal distribution table, we find that the area/proportion between -1 and 2 standard deviations is approximately 0.818.

Therefore, the probability that a randomly selected male from the study has a height between 63 and 87 inches is approximately 0.818 (or 81.8%).

Note: Using the 68-95-99.7 rule is an approximation, and the exact probability can be calculated using the cumulative distribution function of the normal distribution. However, for most practical purposes, the 68-95-99.7 rule provides a good estimate.