Question

Step right up! Refer to Exercise $22$ Suppose that the distances from the tops of the students’ heads to the ground are converted from inches to feet (note that $12in.=1ft$).

a. What shape would the resulting distribution have? Explain your answer.

b. Find the mean of the distribution of distance in feet.

c. Find the standard deviation of the distribution of distance in feet.

Short answer

Part (a) The distribution is skewed to right with a single peak.

Part (b) Mean of the distribution of distance is $6.083$feet.

Part (c) Standard deviation of the distribution of the distance is $0.3575$ feet.

Step-by-step solution

Part (a) Step 1: Given information

$12in.=1ft$

Every distance value in inches must be reduced by $12$ to convert from inches to feet.

Part (a) Step 2: Explanation

We discovered that the distribution of distances was skewed to the right, with a single peak. When every value in the data is split by $12$the form of the distribution remains unchanged since the associations between each data pair remain unchanged. As a result, the form of the distance distribution, in this case, is also skewed to the right, with a single peak.

Part (b) Step 1: Calculation

We discovered that the mean of the distance distribution was $73$inches. We must divide the mean in inches by $12$to translate the mean of the distribution of distances from inches to feet.

$\overline{x}=\frac{73}{12}=6.083ft.$

Thus,

The mean of the distribution of distance in feet is $6.083$ feet.

Part (c) Step 1: Calculation

Standard deviation, $s_x=4.29in.$

We discovered that the standard deviation of the distance distribution was $4.29$inches. We must divide the standard deviation in inches by $12$to convert the standard deviation of a distribution of lengths from inches to feet.

$s_x=\frac{4.29}{12}=0.3575ft.$

Thus,

The standard deviation of the distribution of distance in feet is $0.3575$ feet.