Question

Measure up Clarence measures the diameter of each tennis ball in a bag with a standard ruler. Unfortunately, he uses the ruler incorrectly so that each of his

measurements is $0.2$inches too large. Clarence’s data had a mean of $3.2$inches and a standard deviation of $0.1$ inches. Find the mean and standard deviation of the corrected measurements in centimeters (recall that 1 inch = $2.54$ cm).

Short answer

The mean is $7.62cm$and the median is $0.254cm$

Step-by-step solution

Step 1. Given

C's measurements are all $0.2$inches in length. The average is $3.2$ inches.

$0.1$ inch standard deviation.

Step 2. Concept

The sample standard deviation is measured in the same units as the original data.

Step 3. Calculation

With a standard ruler, the diameter of each tennis ball in a bag was measured, yielding a mean of = 3.2 inches and a standard deviation of = 0.1 inches. It's also been revealed that the ruler was off, thus each measurement is 0.2 inches off. Convert inches to centimeters according to the requirements using the formula inches = 2.54 cm. ............. (1)

To convert a measurement from inches to centimeters, remove 0.2 inches from the mean value and then convert the measurement to centimeters. As a result, each tennis ball's new mean diameter is determined as follows: $$\begin{gathered} {\mu }_{new}=(3.2-0.2)inches \\ =(3.2-0.2)2.54cm \\ =7.62cm \end{gathered}$$

The standard deviation is unaffected by the change in location, thus converting the same measurement to centimeters with the same standard deviation. As a result, for each tennis ball, the new standard deviation of diameter is determined as follows: $$\begin{gathered} {\sigma }_{new}=0.1inch \\ =0.1\times 2.54cm \\ =0.254cm \end{gathered}$$

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