Question

an organization accepts donations of hair longer than 10 inches to be made into wigs for cancer patients. to help out, tabitha got 19 of her classmates to join her in donating their ponytails to the organization. the table shows the lengths, in inches, of the girls ponytails.

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$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \\hline 12 & 15 & 14 & 13.25 & 15.25 & 14.25 & 12.25 & 13 & 12.25 & 15.75 \\\\ \\hline 10 & 11 & 15 & 12 & 11.5 & 12 & 11.75 & 11.75 & 13.25 & 13.25 \\\\ \\hline \\end{array}$$

q. which of the following statements about the measures of center for the data is true?

a. the value 15.75 inches increases the median but has no effect on the mean.
b. the value 15.75 inches has no effect on the mode.
c. the value 10 inches has no effect on the median.
d. the value 10 inches has no effect on the mean.

Explanation:

List and order the data set

Using the Data Ordering knowledge point, we first list all the given ponytail lengths from the table. There are 20 values in total (Tabitha plus her 19 classmates):

$$ 12, 15, 14, 13.25, 15.25, 14.25, 12.25, 13, 12.25, 15.75, 10, 11, 15, 12, 11.5, 12, 11.75, 11.75, 13.25, 13.25 $$

Sorting these 20 values in ascending order:

$$ 10, 11, 11.5, 11.75, 11.75, 12, 12, 12, 12.25, 12.25, 13, 13.25, 13.25, 13.25, 14, 14.25, 15, 15, 15.25, 15.75 $$

Calculate the median of the data set

Using the Median of Data Set knowledge point, since there are \(n = 20\) values (an even number), the median is the average of the 10th and 11th values in the ordered list:

• 10th value: \(12.25\)
• 11th value: \(13\)

$$ \text{Median} = \frac{12.25 + 13}{2} = 12.625 $$

Analyze the effect of removing the value 10

If we remove the value \(10\) (since the organization only accepts hair longer than 10 inches, we look at the effect of this boundary value):
The new ordered list of 19 values is:

$$ 11, 11.5, 11.75, 11.75, 12, 12, 12, 12.25, 12.25, 13, 13.25, 13.25, 13.25, 14, 14.25, 15, 15, 15.25, 15.75 $$

With \(n = 19\) values, the median is the 10th value:

• 10th value: \(13\)

$$ \text{New Median} = 13 $$

Since the median changes from \(12.625\) to \(13\), the value \(10\) does have an effect on the median.

Analyze the effect of removing the value 15.75

If we remove the maximum value \(15.75\), the new ordered list of 19 values is:

$$ 10, 11, 11.5, 11.75, 11.75, 12, 12, 12, 12.25, 12.25, 13, 13.25, 13.25, 13.25, 14, 14.25, 15, 15, 15.25 $$

With \(n = 19\) values, the median is the 10th value:

• 10th value: \(12.25\)

$$ \text{New Median} = 12.25 $$

The mean is also affected because removing any single value changes the sum and the count.
Now let's check the mode of the original data set:

• The value \(12\) appears 3 times.
• The value \(13.25\) appears 3 times.

Thus, the original data set is bimodal with modes \(12\) and \(13.25\).
If we remove \(15.75\), the frequencies of \(12\) and \(13.25\) remain 3, so the mode is unchanged. Therefore, the value \(15.75\) has no effect on the mode.

Answer:

• (A) The value 15.75 inches increases the median but has no effect on the mean.
• (B) The value 15.75 inches has no effect on the mode. (Correct answer)
• (C) The value 10 inches has no effect on the median.
• (D) The value 10 inches has no effect on the mean.