Question

Suppose a random sample of adult women has a sample mean height of x¯=64.3 inches, with a sample standard deviation of s=2.4 inches. Since height distribution are generally symmetric and bell-shaped, we can apply the Empirical Rule. Between what two heights are approximately 99.7% of the data? Round your answers to the nearest tenth.

Answer 1

Answer:The Empirical Rule, also known as the 68-95-99.7 Rule, states that for a normal distribution:

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

About 95% falls within two standard deviations.

About 99.7% falls within three standard deviations.

In this case, you are interested in the interval within which approximately 99.7% of the data falls. Since the data is approximately normally distributed, you can use the sample mean (

�

ˉ

x

ˉ

) and the sample standard deviation (

�

s) to estimate this interval.

So, for approximately 99.7% of the data, you would go three standard deviations above and below the mean. Therefore, the interval is given by:

�

ˉ

−

3

�

to

�

ˉ

+

3

�

x

ˉ

−3s to

x

ˉ

+3s

Substitute the given values:

64.3

−

3

×

2.4

to

64.3

+

3

×

2.4

64.3−3×2.4 to 64.3+3×2.4

Calculate:

64.3

−

7.2

to

64.3

+

7.2

64.3−7.2 to 64.3+7.2

This gives the interval:

57.1

to

71.5

57.1 to 71.5

So, approximately 99.7% of the data falls between 57.1 inches and 71.5 inches.

Explanation: