Sure! Let's solve this step-by-step.
We know that women's heights are normally distributed with a mean (\(\mu\)) of 63.6 inches and a standard deviation (\(\sigma\)) of 2.5 inches.
### Part (a): The probability that a randomly selected woman will be taller than 62 inches.
1. Calculate the Z-score for 62 inches:
The Z-score formula is:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
For \( X = 62 \), \(\mu = 63.6\), and \(\sigma = 2.5\):
[tex]\[ Z_a = \frac{62 - 63.6}{2.5} = -0.64 \][/tex]
2. Find the probability corresponding to the Z-score:
The Z-score of -0.64 corresponds to the cumulative probability (area to the left of Z) which can be found using standard normal distribution tables or statistical software. This cumulative probability, \( P(Z \leq -0.64) \), is approximately 0.2611.
3. Calculate the probability of being taller than 62 inches:
[tex]\[ P(X > 62) = 1 - P(Z \leq -0.64) = 1 - 0.2611 = 0.7389 \][/tex]
Therefore, the probability that a randomly selected woman will be taller than 62 inches is approximately 0.7389 (or 73.89%).
### Part (b): The probability that a randomly selected woman will be between 64 and 71 inches tall.
1. Calculate the Z-scores for 64 inches and 71 inches:
For \(X = 64\):
[tex]\[ Z_{lower} = \frac{64 - 63.6}{2.5} = 0.16 \][/tex]
For \(X = 71\):
[tex]\[ Z_{upper} = \frac{71 - 63.6}{2.5} = 2.96 \][/tex]
2. Find the cumulative probabilities corresponding to these Z-scores:
\( P(Z \leq 0.16) \), the cumulative probability for \( Z = 0.16 \), is approximately 0.5636.
\( P(Z \leq 2.96) \), the cumulative probability for \( Z = 2.96 \), is approximately 0.9985.
3. Calculate the probability of being between 64 and 71 inches:
[tex]\[ P(64 \leq X \leq 71) = P(Z \leq 2.96) - P(Z \leq 0.16) = 0.9985 - 0.5636 = 0.4349 \][/tex]
Therefore, the probability that a randomly selected woman will be between 64 and 71 inches tall is approximately 0.4349 (or 43.49%).
In conclusion:
- The probability that a woman will be taller than 62 inches is approximately 0.7389.
- The probability that a woman will be between 64 and 71 inches tall is approximately 0.4349.
