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
According to the Empirical Rule, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
To find the probability that a man will have a shoe size between 13 and 14.5, we first need to determine how many standard deviations away these sizes are from the mean:
1. For size 13:
\( z = \frac{13 - 10}{1.5} = 2 \)
2. For size 14.5:
\( z = \frac{14.5 - 10}{1.5} = 3 \)
Since one standard deviation is about 68% of the data and two standard deviations cover 95%, we can reasonably estimate that the probability of a shoe size falling between 13 and 14.5 to be 95%.
is this right
I apologize for the mistake 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 need to first convert these sizes to z-scores:
1. For size 13:
\( z = \frac{13 - 10}{1.5} = 2 \)
2. For size 14.5:
\( z = \frac{14.5 - 10}{1.5} = 3 \)
Next, we look up the z-scores in a standard normal distribution table to find the area under the curve between these z-scores.
For a z-score of 2, the area to the left is approximately 0.9772.
For a z-score of 3, the area to the left is approximately 0.9987.
To find the probability that a man will have a shoe size between 13 and 14.5, we subtract the probability of having a shoe size less than 13 (0.9772) from the probability of having a shoe size less than 14.5 (0.9987):
\(P(13 < x < 14.5) = P(x<14.5) - P(x<13) = 0.9987 - 0.9772 = 0.0215 \)
Therefore, the 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 appreciate your understanding.