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.
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
The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ).
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Given:
- Mean (μ) = 10
- Standard Deviation (σ) = 1.5
To use the Empirical Rule to find the probability of a man having a shoe size between 13 and 14.5, we first calculate how many standard deviations away from the mean each of these sizes is.
1. For a shoe size of 13:
\[
z = \frac{X - μ}{σ} = \frac{13 - 10}{1.5} = \frac{3}{1.5} = 2
\]
2. For a shoe size of 14.5:
\[
z = \frac{X - μ}{σ} = \frac{14.5 - 10}{1.5} = \frac{4.5}{1.5} = 3
\]
Now we have:
- Shoe size 13 corresponds to \( z = 2 \)
- Shoe size 14.5 corresponds to \( z = 3 \)
According to the Empirical Rule:
- Approximately 95% of the data falls within 2 standard deviations from the mean (between \( μ - 2σ \) and \( μ + 2σ \)).
- Approximately 99.7% of the data falls within 3 standard deviations from the mean (between \( μ - 3σ \) and \( μ + 3σ \)).
Since 95% covers the range between shoe sizes \( 7 \) (2 standard deviations below 10) and \( 13 \) (2 standard deviations above 10), we are interested in the range from a z-score of 2 to a z-score of 3.
To find the area (or probability) between \( z = 2 \) and \( z = 3 \):
1. The probability of being below z = 3 is 99.7% (or 0.997).
2. The probability of being below z = 2 is 97.7% (or 0.977).
Now, to find the probability between z = 2 and z = 3:
\[
P(2 < z < 3) = P(z < 3) - P(z < 2) = 0.997 - 0.977 = 0.020
\]
Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately **0.020**, or **2%**.
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ).
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Given:
- Mean (μ) = 10
- Standard Deviation (σ) = 1.5
To use the Empirical Rule to find the probability of a man having a shoe size between 13 and 14.5, we first calculate how many standard deviations away from the mean each of these sizes is.
1. For a shoe size of 13:
\[
z = \frac{X - μ}{σ} = \frac{13 - 10}{1.5} = \frac{3}{1.5} = 2
\]
2. For a shoe size of 14.5:
\[
z = \frac{X - μ}{σ} = \frac{14.5 - 10}{1.5} = \frac{4.5}{1.5} = 3
\]
Now we have:
- Shoe size 13 corresponds to \( z = 2 \)
- Shoe size 14.5 corresponds to \( z = 3 \)
According to the Empirical Rule:
- Approximately 95% of the data falls within 2 standard deviations from the mean (between \( μ - 2σ \) and \( μ + 2σ \)).
- Approximately 99.7% of the data falls within 3 standard deviations from the mean (between \( μ - 3σ \) and \( μ + 3σ \)).
Since 95% covers the range between shoe sizes \( 7 \) (2 standard deviations below 10) and \( 13 \) (2 standard deviations above 10), we are interested in the range from a z-score of 2 to a z-score of 3.
To find the area (or probability) between \( z = 2 \) and \( z = 3 \):
1. The probability of being below z = 3 is 99.7% (or 0.997).
2. The probability of being below z = 2 is 97.7% (or 0.977).
Now, to find the probability between z = 2 and z = 3:
\[
P(2 < z < 3) = P(z < 3) - P(z < 2) = 0.997 - 0.977 = 0.020
\]
Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately **0.020**, or **2%**.