Chapter 2.1, Problem 24E

To determine

To find: the mean and standard deviation of the corrected measurements

Answer to Problem 24E

The mean is 7.62cm and the median is 0.254cm

Explanation of Solution

Given:

Each of C’s measurement is 0.2 inches large.

Mean = 3.2 inches

Standard deviation = 0.1 inches

Calculation:

The diameter of each tennis ball in a bag was measured with a standard ruler and obtained a mean of $\mu =3.2$ inches and a standard deviation of $\sigma =0.1$ inches. Also it is informed that the ruler was incorrect, so that each of the measurements is 0.2 inches large. As per the requirement, convert inches into centimeter using the formula linch = 2.54 cm...... (1)

To get a correct measurement in centimeters, it is needed to subtract the 0.2 inches from the mean value and then transform the same measurement to centimeters. Thus the new mean diameter of each tennis ball is calculated as follows:

  $\begin{array}{l}{\mu }_{new}=(3.2-0.2)\text{inches}\\ =(3.2-0.2)\text{2}\text{.54cm}\\ =7.62\text{cm}\end{array}$

The change in location does not affect the standard deviation so the standard deviation remains same and convert the same measurement to centimeters. Thus the new standard deviation of diameter for each tennis ball is calculated as follows:

  $\begin{array}{l}{\sigma }_{new}=0.1\text{ inch}\\ \text{=0}\text{.1}\times \text{2}\text{.54 cm}\\ \text{=0}\text{.254 cm}\end{array}$

Conclusion:

Therefore, the mean is 7.62cm and the median is 0.254cm