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
Answers
GPT-4o mini
To use the Empirical Rule, also known as the 68-95-99.7 Rule, we need to understand how the normal distribution works in terms of standard deviations from the mean.
In your case:
- Mean (μ) = 10
- Standard Deviation (σ) = 1.5
According to the Empirical Rule:
1. About 68% of the data falls within ±1 standard deviation from the mean.
2. About 95% of the data falls within ±2 standard deviations from the mean.
3. About 99.7% of the data falls within ±3 standard deviations from the mean.
### Calculation of the Z-scores
First, we need to determine how many standard deviations away from the mean the values 13 and 14.5 are.
**For Shoe Size = 13:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{13 - 10}{1.5} = \frac{3}{1.5} = 2
\]
**For Shoe Size = 14.5:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{14.5 - 10}{1.5} = \frac{4.5}{1.5} = 3
\]
### Range of Z-scores
Thus, the Z-scores for shoe sizes 13 and 14.5 are:
- 13 corresponds to \( Z = 2 \)
- 14.5 corresponds to \( Z = 3 \)
### Probability between Z = 2 and Z = 3
Using the standard normal distribution:
- Approximately 95% of the data lies within ±2 standard deviations from the mean (Z = -2 to Z = 2).
- Approximately 99.7% of the data lies within ±3 standard deviations from the mean (Z = -3 to Z = 3).
To find the probability between Z = 2 and Z = 3:
1. Probability up to Z = 2 (from the Z-table):
- This is about 0.9750 or 97.5%.
2. Probability up to Z = 3 (from the Z-table):
- This is about 0.9987 or 99.87%.
### Finding the Probability between Z = 2 and Z = 3
Now, to find the probability of a shoe size between 13 and 14.5:
\[
P(13 < X < 14.5) = P(Z < 3) - P(Z < 2) = 0.9987 - 0.9750 = 0.0237
\]
Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately **0.0237**, or **2.37%**.
In your case:
- Mean (μ) = 10
- Standard Deviation (σ) = 1.5
According to the Empirical Rule:
1. About 68% of the data falls within ±1 standard deviation from the mean.
2. About 95% of the data falls within ±2 standard deviations from the mean.
3. About 99.7% of the data falls within ±3 standard deviations from the mean.
### Calculation of the Z-scores
First, we need to determine how many standard deviations away from the mean the values 13 and 14.5 are.
**For Shoe Size = 13:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{13 - 10}{1.5} = \frac{3}{1.5} = 2
\]
**For Shoe Size = 14.5:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{14.5 - 10}{1.5} = \frac{4.5}{1.5} = 3
\]
### Range of Z-scores
Thus, the Z-scores for shoe sizes 13 and 14.5 are:
- 13 corresponds to \( Z = 2 \)
- 14.5 corresponds to \( Z = 3 \)
### Probability between Z = 2 and Z = 3
Using the standard normal distribution:
- Approximately 95% of the data lies within ±2 standard deviations from the mean (Z = -2 to Z = 2).
- Approximately 99.7% of the data lies within ±3 standard deviations from the mean (Z = -3 to Z = 3).
To find the probability between Z = 2 and Z = 3:
1. Probability up to Z = 2 (from the Z-table):
- This is about 0.9750 or 97.5%.
2. Probability up to Z = 3 (from the Z-table):
- This is about 0.9987 or 99.87%.
### Finding the Probability between Z = 2 and Z = 3
Now, to find the probability of a shoe size between 13 and 14.5:
\[
P(13 < X < 14.5) = P(Z < 3) - P(Z < 2) = 0.9987 - 0.9750 = 0.0237
\]
Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately **0.0237**, or **2.37%**.