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.
To apply the Empirical Rule, we need to calculate the z-scores for shoe sizes 13 and 14.5.
For shoe size 13:
z = (13 - 10) / 1.5 = 2
For shoe size 14.5:
z = (14.5 - 10) / 1.5 = 3
Using the Empirical Rule, we know that:
- Approximately 68% of the data falls within 1 standard deviation of the mean
- 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
Since shoe size 13 is 2 standard deviations above the mean, and shoe size 14.5 is 3 standard deviations above the mean, we can expect the probability of a man having a shoe size between 13 and 14.5 to be approximately:
P(13 < shoe size < 14.5) = P(10 < shoe size < 13) + P(10 < shoe size < 14.5)
= 95% + 99.7%
= 194.7%
Therefore, the probability that a man will have a shoe size between 13 and 14.5 is approximately 94.7%.
is this right?
I apologize for the error in my previous response. Let's correct the calculation.
To find the probability that a man will have a shoe size between 13 and 14.5, we first need to find the z-scores for both shoe sizes using the formula:
z = (x - μ) / σ
For shoe size 13:
z1 = (13 - 10) / 1.5 = 2
For shoe size 14.5:
z2 = (14.5 - 10) / 1.5 = 3
Next, we need to find the probabilities associated with these z-scores using a standard normal distribution table or calculator.
The probability associated with z = 2 is approximately 0.9772, and the probability associated with z = 3 is approximately 0.9987.
To find the probability that a man will have a shoe size between 13 and 14.5, we need to subtract the probability of z = 3 from the probability of z = 2:
P(13 < shoe size < 14.5) = P(2 < z < 3) = P(z < 3) - P(z < 2)
= 0.9987 - 0.9772
= 0.0215
Therefore, the correct probability that a man will have a shoe size between 13 and 14.5 is approximately 2.15%. Thank you for pointing out the error, and I hope this clarifies the calculation.