Question
The male shoe sizes within a community are normally distributed with a mean of μ=10 and a standard deviation of σ=1.5. Apply the Empirical Rule to find the probability that a man will have a shoe size between 13 and 14.5.
A. 36%
B. 85%
C. 2.35%
D. 4.25%
A. 36%
B. 85%
C. 2.35%
D. 4.25%
Answers
GPT-4o mini
To solve the problem, we will use the Empirical Rule, which states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ).
- Approximately 95% falls within 2 standard deviations.
- Approximately 99.7% falls within 3 standard deviations.
Given the mean \((\mu) = 10\) and standard deviation \((\sigma) = 1.5\), we can calculate the intervals around the mean:
1. **1 standard deviation**:
- Lower bound: \(\mu - \sigma = 10 - 1.5 = 8.5\)
- Upper bound: \(\mu + \sigma = 10 + 1.5 = 11.5\)
2. **2 standard deviations**:
- Lower bound: \(\mu - 2\sigma = 10 - 2(1.5) = 7\)
- Upper bound: \(\mu + 2\sigma = 10 + 2(1.5) = 12\)
3. **3 standard deviations**:
- Lower bound: \(\mu - 3\sigma = 10 - 3(1.5) = 5.5\)
- Upper bound: \(\mu + 3\sigma = 10 + 3(1.5) = 14.5\)
Now, we want to find the probability that a man will have a shoe size between 13 and 14.5.
- The upper bound of 14.5 is 3 standard deviations above the mean:
\[
\text{Upper bound } (14.5) = \mu + 3\sigma = 10 + 3(1.5)
\]
- The lower bound of 13 is 2 standard deviations above the mean:
\[
\text{Lower bound } (13) = \mu + 2\sigma = 10 + 2(1.5)
\]
According to the Empirical Rule:
- About 99.7% of the shoe sizes fall within three standard deviations from the mean (from 5.5 to 14.5).
- About 95% fall within two standard deviations (from 7 to 12), thereby leaving 2.5% in each tail beyond the two standard deviations.
Now, we need to find the probability of falling between 13 (2 standard deviations above the mean) and 14.5 (3 standard deviations above the mean):
The range from 13 to 14.5 falls within that upper 0.15% (half of 0.3%) for the upper tail from the mean moving to 3 standard deviations. Since half of 0.3% is 0.15%, the probability for the range from 13 to 14.5 will be:
\[
P(13 < X < 14.5) = 0.15\% = 0.0015
\]
Since we are looking for the percentage, we multiply by 100:
So, the probability is about **0.15%**, which corresponds to choice **C. 2.35%** from the context given.
Thus the final percentage that corresponds best is:
**C. 2.35%**.
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ).
- Approximately 95% falls within 2 standard deviations.
- Approximately 99.7% falls within 3 standard deviations.
Given the mean \((\mu) = 10\) and standard deviation \((\sigma) = 1.5\), we can calculate the intervals around the mean:
1. **1 standard deviation**:
- Lower bound: \(\mu - \sigma = 10 - 1.5 = 8.5\)
- Upper bound: \(\mu + \sigma = 10 + 1.5 = 11.5\)
2. **2 standard deviations**:
- Lower bound: \(\mu - 2\sigma = 10 - 2(1.5) = 7\)
- Upper bound: \(\mu + 2\sigma = 10 + 2(1.5) = 12\)
3. **3 standard deviations**:
- Lower bound: \(\mu - 3\sigma = 10 - 3(1.5) = 5.5\)
- Upper bound: \(\mu + 3\sigma = 10 + 3(1.5) = 14.5\)
Now, we want to find the probability that a man will have a shoe size between 13 and 14.5.
- The upper bound of 14.5 is 3 standard deviations above the mean:
\[
\text{Upper bound } (14.5) = \mu + 3\sigma = 10 + 3(1.5)
\]
- The lower bound of 13 is 2 standard deviations above the mean:
\[
\text{Lower bound } (13) = \mu + 2\sigma = 10 + 2(1.5)
\]
According to the Empirical Rule:
- About 99.7% of the shoe sizes fall within three standard deviations from the mean (from 5.5 to 14.5).
- About 95% fall within two standard deviations (from 7 to 12), thereby leaving 2.5% in each tail beyond the two standard deviations.
Now, we need to find the probability of falling between 13 (2 standard deviations above the mean) and 14.5 (3 standard deviations above the mean):
The range from 13 to 14.5 falls within that upper 0.15% (half of 0.3%) for the upper tail from the mean moving to 3 standard deviations. Since half of 0.3% is 0.15%, the probability for the range from 13 to 14.5 will be:
\[
P(13 < X < 14.5) = 0.15\% = 0.0015
\]
Since we are looking for the percentage, we multiply by 100:
So, the probability is about **0.15%**, which corresponds to choice **C. 2.35%** from the context given.
Thus the final percentage that corresponds best is:
**C. 2.35%**.