Question

Birth Weights Based on Data Set 4 “Births” in Appendix B, birth weights are normally distributed with a mean of 3152.0 g and a standard deviation of 693.4 g.

a. What are the values of the mean and standard deviation after converting all birth weights to z scores using $z=\frac{x-\mu }{\sigma }$?

b. The original birth weights are in grams. What are the units of the corresponding z scores?

Short answer

a. Mean: 0

Standard Deviation: 1

b. The z-score values do not have any units. They are just numbers.

Step-by-step solution

Given information

The birth weights are normally distributed with a mean equal to 3152.0 grams and a standard deviation equal to 693.4 grams.

z-scores

a.

It is given that the birth weights follow normal distribution with mean equal to 3152.0 grams and standard deviation equal to 693.4 grams.

These birth weights are then converted to z-scores by subtracting the mean value from them and then dividing them by the standard deviation.

The obtained z-scores of the birth weights now follow the normal distribution with mean value equal to 0 and standard deviation equal to 1.

Thus, the mean value is equal to 0 and the standard deviation is equal to 1.

b.

The conversion of data values to z-scores has the following expression:

$z=\frac{x-\mu }{\sigma }$

Here, both the denominator and the numerator have the same units (grams). Thus, after division the resulting z-scores do not have any units.

The z-score values do not have any units. They are mere numbers.