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 $0.2inch$too large. Clarence’s data had a mean of $3.2inches$and a standard deviation of $0.1inch$ (Recall that $1in.=2.54cm$)

a. Find the mean of the corrected measurements in centimeters.

b. Calculate the standard deviation of the corrected measurements in centimeters.

Short answer

Part (a) Mean of the corrected measurements is $7.62cm$

Part (b) Standard deviation of the corrected measurements is $0.254cm$

Step-by-step solution

Part (a) Step 1: Given information and Explanation

$1in.=2.54cm$

Mean,

$\overline{x_{in}}=3.2inches$

Standard deviation,

${SD}_{in}=0.1inches$

Every data value must be multiplied by $2.54$in order to convert from inches to centimeters.

We must reduce the measurement (or data value) by $0.2$inches for corrected measurements since the measurement was $0.2$inches too large.

Because the center of the measurements is represented by the mean.

When every data value is multiplied by $2.54$then the center of the distribution is also multiplied by $2.54$

Similarly,

When every data value is decreased by $0.2inches$then the center of the distribution is also decreased by $0.2inches$

Then

The new mean of the corrected measurements in centimeters is the mean of the uncorrected measurements in inches decreased by $0.2inches$

multiplied by $2.54$

New mean,

$\overline{x_{cm}}=(\overline{x_{in}}-0.2).2.54=(3.2-0.2)\cdot 2.54=\left(3\right)\cdot 2.54=7.62cm$

Thus,

The mean of the corrected measurements in centimeters is $7.62cm$

Part (b) Step 1: Calculation

Standard deviation, ${SD}_{in.}=0.1inches$

Every data value must be multiplied by $2.54$ in order to convert from inches to centimeters. We must reduce the measurement (or data value) by $0.2$ inches for corrected measurements since the measurement was $0.2$ inches too large. Because the standard deviation is a measure of variance.

When each data value is multiplied by $2.54$, the measurement variability is also multiplied by $2.54$ And the variability of the measurements will not be changed if each data value is reduced by $0.2inches$

Then

The corrected measurement's new standard deviation in centimeters equals the uncorrected measurement's standard deviation multiplied by $2.54$

New standard deviation,

${SD}_{cm.}=2.54\left({SD}_{in.}\right)=2.54(0.1)=0.254cm$

Thus,

The standard deviation of the corrected measurements in centimeters is $0.254cm$