To solve this problem, you'll first need to understand how to use the properties of the normal distribution to find the percentage of women whose weights fall between the given limits.
Given:
- Mean weight of women, [tex]\( \mu = 176.1 \)[/tex] lbs
- Standard deviation of women's weights, [tex]\( \sigma = 40.8 \)[/tex] lbs
- Lower weight limit, [tex]\( L = 136.7 \)[/tex] lbs
- Upper weight limit, [tex]\( U = 202.0 \)[/tex] lbs
We need to find the percentage of women whose weights are between 136.7 lbs and 202.0 lbs.
### Step-by-Step Solution:
1. Calculate the Z-scores for the weight limits:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
Here, [tex]\(X\)[/tex] is the value of interest (either the lower or upper limit), [tex]\(\mu\)[/tex] is the mean, and [tex]\(\sigma\)[/tex] is the standard deviation.
- For the lower limit:
[tex]\[
Z_{\text{lower}} = \frac{136.7 - 176.1}{40.8} = \frac{-39.4}{40.8} \approx -0.97
\][/tex]
- For the upper limit:
[tex]\[
Z_{\text{upper}} = \frac{202.0 - 176.1}{40.8} = \frac{25.9}{40.8} \approx 0.63
\][/tex]
2. Find the cumulative probability for these Z-scores:
The cumulative distribution function (CDF) for a standard normal distribution gives the probability that a standard normal variable is less than or equal to a given value [tex]\(Z\)[/tex].
- [tex]\(P(Z \leq -0.97)\)[/tex] can be found using Z-tables or a standard normal distribution calculator:
[tex]\[
P(Z \leq -0.97) \approx 0.166
\][/tex]
- [tex]\(P(Z \leq 0.63)\)[/tex]:
[tex]\[
P(Z \leq 0.63) \approx 0.735
\][/tex]
3. Calculate the percentage of women within the given weight limits:
The percentage of women with weights between 136.7 lbs and 202.0 lbs is:
[tex]\[
P(L \leq W \leq U) = P(Z \leq 0.63) - P(Z \leq -0.97)
\][/tex]
[tex]\[
\approx 0.735 - 0.166 = 0.569
\][/tex]
To express this as a percentage:
[tex]\[
0.569 \times 100\% = 56.9\%
\][/tex]
Therefore, approximately [tex]\(56.90\%\)[/tex] of women have weights that are within the specified limits.
### Are many women excluded with those specifications?
Given that only about 56.90% of women fall within the weight range for the aircraft ejection seats, it means that approximately 43.10% (100% - 56.90%) of women are excluded. Yes, indeed, a significant proportion of women are excluded based on these specifications.
