Question

the distances (in yards) for nine holes of a golf course are listed below.
335 392 408 523 147 503 177 377 360
complete parts (a) through (d) below.

(a) find the mean and median of the data.
the mean of the data is 358.
the median of the data is 377.

(b) convert the distances to feet. then rework part (a).
the mean of the data in feet is 1074.
the median of the data in feet is 1131.

(c) compare the measures you found in part (b) with those found in part (a). what do you notice?
a. the difference between the mean in part (b) and the mean in part (a) is equal to the similar difference between the medians.
b. the mean and median in part (b) are three times the mean and median in part (a).
c. the mean and median in part (b) are not the mean and median in part (a) expressed in feet.
d. the ratio of the mean in part (b) to the mean in part (a) is not equal to the similar ratio of the medians.

(d) use your results from part (c) to explain how to quickly find the mean and median of the given data set when the distances are converted to inches.
a. if you multiply the mean and median from part (a) by 3, you will get the mean and median of the data set in inches.
b. if you multiply the mean and median from part (a) by 12, you will get the mean and median of the data set in inches.
c. if you convert all the distances to feet and then rework part (a), it will be the quickest way to obtain the mean and median in inches.
d. if you multiply the mean and median from part (b) by 12, you will get the mean and median of the data set in inches.

Explanation:

Calculate mean and median in yards

Using the Mean Calculation and Median Calculation knowledge points
The given dataset in yards is:

$$ \{335, 392, 408, 523, 147, 503, 177, 377, 360\} $$

Sum of the data:

$$ \sum x = 335 + 392 + 408 + 523 + 147 + 503 + 177 + 377 + 360 = 3222 $$

Mean in yards:

$$ \bar{x} = \frac{3222}{9} = 358 $$

Sorted dataset in yards:

$$ \{147, 177, 335, 360, 377, 392, 408, 503, 523\} $$

Median (the 5th value):

$$ \text{Median} = 377 $$

Convert to feet and recalculate

Using the Linear Transformation of Data knowledge point
Since \(1\text{ yard} = 3\text{ feet}\), we multiply each value by 3.
Mean in feet:

$$ \bar{x}_{\text{feet}} = 358 \times 3 = 1074 $$

Median in feet:

$$ \text{Median}_{\text{feet}} = 377 \times 3 = 1131 $$

Compare measures

Using the Linear Transformation of Data knowledge point
Comparing the values:

$$ \frac{1074}{358} = 3 \quad \text{and} \quad \frac{1131}{377} = 3 $$

This shows that the mean and median in feet are three times the mean and median in yards. This corresponds to option B.

Generalize to inches conversion

Using the Linear Transformation of Data knowledge point
To convert yards directly to inches, we multiply by 36 (since \(1\text{ yard} = 36\text{ inches}\)).
Alternatively, to convert feet to inches, we multiply by 12 (since \(1\text{ foot} = 12\text{ inches}\)).
Option D states: "If you multiply the mean and median from part (b) by 12, you will get the mean and median of the data set in inches." This is correct because part (b) measures are in feet.

Answer:

Question 1

The mean of the data is 358.
The median of the data is 377.

Question 2

The mean of the data in feet is 1074.
The median of the data in feet is 1131.

Question 3

• A. The difference between the mean in part (b) and the mean in part (a) is equal to the similar difference between the medians.
• B. The mean and median in part (b) are three times the mean and median in part (a). (Correct answer)
• C. The mean and median in part (b) are not the mean and median in part (a) expressed in feet.
• D. The ratio of the mean in part (b) to the mean in part (a) is not equal to the similar ratio of the medians.

Question 4

• A. If you multiply the mean and median from part (a) by 3, you will get the mean and median of the data set in inches.
• B. If you multiply the mean and median from part (a) by 12, you will get the mean and median of the data set in inches.
• C. If you convert all the distances to feet and then rework part (a), it will be the quickest way to obtain the mean and the median in inches.
• D. If you multiply the mean and median from part (b) by 12, you will get the mean and median of the data set in inches. (Correct answer)