Questions: Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The standard deviation for the heights of a group of 5-year-old children is smaller than the standard deviation for the heights of a group of children who range in age from 3 to 15. Choose the correct answer below. A. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. B. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15. C. The statement makes sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. D. The statement makes sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15.

Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The standard deviation for the heights of a group of 5-year-old children is smaller than the standard deviation for the heights of a group of children who range in age from 3 to 15. Choose the correct answer below. A. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. B. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15. C. The statement makes sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. D. The statement makes sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15.
Transcript text: Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The standard deviation for the heights of a group of 5-year-old children is smaller than the standard deviation for the heights of a group of children who range in age from 3 to 15. Choose the correct answer below. A. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. B. The statement does not make sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15. C. The statement makes sense because the range of data for the heights of a group of 5-year-old children is smaller than the range of data for the heights of a group of children who range in age from 3 to 15. D. The statement makes sense because the range of data for the heights of a group of 5-year-old children is larger than the range of data for the heights of a group of children who range in age from 3 to 15.
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Solution

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Solution Steps

Step 1: Calculate the Mean and Standard Deviation for 5-Year-Old Children

For the heights of the group of 5-year-old children, we have the following calculations:

  • Mean \( \mu \): \[ \mu = \frac{\sum x_i}{n} = \frac{1065}{10} = 106.5 \]

  • Variance \( \sigma^2 \): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 1.17 \]

  • Standard Deviation \( \sigma \): \[ \sigma = \sqrt{1.17} \approx 1.08 \]

Thus, the standard deviation for 5-year-old children is \( 1.08 \).

Step 2: Calculate the Mean and Standard Deviation for Children Aged 3 to 15

For the heights of the group of children aged 3 to 15, we have the following calculations:

  • Mean \( \mu \): \[ \mu = \frac{\sum x_i}{n} = \frac{1560}{13} = 120.0 \]

  • Variance \( \sigma^2 \): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 379.17 \]

  • Standard Deviation \( \sigma \): \[ \sigma = \sqrt{379.17} \approx 19.47 \]

Thus, the standard deviation for children aged 3 to 15 is \( 19.47 \).

Step 3: Compare the Standard Deviations

We compare the standard deviations calculated for both groups:

  • Standard Deviation for 5-year-old children: \( 1.08 \)
  • Standard Deviation for children aged 3 to 15: \( 19.47 \)

Since \( 1.08 < 19.47 \), the statement that the standard deviation for the heights of a group of 5-year-old children is smaller than that of the heights of a group of children who range in age from 3 to 15 makes sense.

Final Answer

The answer is \( \boxed{C} \).

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