Question

A study was conducted on shoe sizes of students, reported in European sizes. For the women, the mean size was 38.73 with a standard deviation of 1.75. To convert

European shoe sizes to U.S. sizes for women, use the equation shown below.

USsize = EuroSize x 0.7987 - 22.2

a) What is the mean women's shoe size for these responses in U.S. units?

b) What is the standard deviation in U.S. units?

a) The mean women's shoe size in U.S. units is (8.73.

(Round to two decimal places as needed.)

s

.

b) The standard deviation in U.S. units is

(Round to two decimal places as needed.)

Answer 1

The mean women's shoe size in U.S. units is 8.73.

The standard deviation in U.S. units is 1.40.

Explanation:

For the women, the mean size was 38.73 with a standard deviation of 1.75. This size is expressed in European units.

If we want to convert to US units, we have to use the equation:

`US\, size=EuroSize*0.7987-22.2`

If we use the properties of the expected value, then the mean expressed in US units is:

`Property: E(y)=E(ax+b)=aE(x)+b\\\\\\E(y)=0.7987E(x)-22.2\\\\E(y)=0.7987*38.73-22.2\\\\E(y)=8.73`

To calculate the standard deviation, we use the properties of variance:

`Property: V(y)=V(ax+b)=a^2V(x)\\\\\sigma_y=√(a^2V(x))=a\sigma_x\\\\\sigma_y=0.7987*1.75=1.40`