Joan's finishing time for the Bolder Boulder 10K race was 1.77 standard deviations faster than the women's average for her age group. There were 415 women who ran in her age group. Assuming a normal distribution, how many women ran faster than Joan? (Round down your answer to the nearest whole number.) Number of women

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**Homework: Chapter 7 (Sections 7.1 through 7.4)**

**Joan's finishing time for the Boulder Boulder 10K race was 1.77 standard deviations faster than the women's average for her age group. There were 415 women who ran in her age group. Assuming a normal distribution, how many women ran faster than Joan? (Round down your answer to the nearest whole number.)**

**Number of women:** [ ]

**Explanation:**

Joan's performance can be understood using the properties of the normal distribution. Her time being 1.77 standard deviations faster means we need to find the area under the normal curve to the right of a z-value of 1.77. This area represents the proportion of women who finished faster than Joan.

Typically, the z-value of 1.77 corresponds to the 96th percentile of a standard normal distribution, meaning the remaining 4% of the runners (those who are faster) lie to the right of this z-value.

To find the number of women faster than Joan:
\[ \text{Number of women faster than Joan} = \text{Total women} \times \text{Percentage faster than Joan} \]
\[ = 415 \times 0.04 \]
\[ = 16.6 \]

Rounding down to the nearest whole number, we get 16 women who ran faster than Joan.
Transcribed Image Text:**Homework: Chapter 7 (Sections 7.1 through 7.4)** **Joan's finishing time for the Boulder Boulder 10K race was 1.77 standard deviations faster than the women's average for her age group. There were 415 women who ran in her age group. Assuming a normal distribution, how many women ran faster than Joan? (Round down your answer to the nearest whole number.)** **Number of women:** [ ] **Explanation:** Joan's performance can be understood using the properties of the normal distribution. Her time being 1.77 standard deviations faster means we need to find the area under the normal curve to the right of a z-value of 1.77. This area represents the proportion of women who finished faster than Joan. Typically, the z-value of 1.77 corresponds to the 96th percentile of a standard normal distribution, meaning the remaining 4% of the runners (those who are faster) lie to the right of this z-value. To find the number of women faster than Joan: \[ \text{Number of women faster than Joan} = \text{Total women} \times \text{Percentage faster than Joan} \] \[ = 415 \times 0.04 \] \[ = 16.6 \] Rounding down to the nearest whole number, we get 16 women who ran faster than Joan.
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