Question

Transformations The heights (in inches) of men listed in Data Set 1 “Body Data” in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population.

a. If 2 inches is added to each height, are the new heights also normally distributed?

b. If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed?

c. Are the logarithms of normally distributed heights also normally distributed?

Short answer

a.If 2 inches is added to each height, then the new heights follow the normal distribution.

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B.If each height is converted from inches to centimeters, then the heights in centimeters are normally distributed.

c. The logarithm of the normally distributed heights does not follow normal distribution.

Step-by-step solution

Given information

The heights (in inches) of men appear to be from a normally distributed population.

Describe the distribution of heights when a constant is added

a. Yes, 2 inches is added to each height, the new heights are normally distributed.

Because the heights follow normal distribution, the shape of the distribution is bell shaped. If2 inches (a constant) is added to each observed height, the heights will remain bell shaped that is normally distributed, with a shift in mean measure (point of symmetry).

Describe the distribution of heights when a constant is multiplied

b. If each height is converted from inches to centimeters, the heights in centimeters would also be normally distributed.

To convert heights from inches to centimeters, multiply original heights by 2.54 (a constant conversion factor).

Due to change in observation, the pattern of heights will remain same as the original heights that is normally distributed. But there would be shift of mean as well as the standard deviation measure.

Describe the distributions of logarithms of normal distribution

c. The logarithms of normally distributed heights are not normally distributed. The distribution of such values is lognormal.

This is because the logarithmic function is not linear and the logarithm will change the shape of the distribution.