Answer :
(D) 82%.
To find the percentage of female college cross-country runners who weigh between 114 pounds and 138 pounds, given a normal distribution with a mean of 122 pounds and a standard deviation of 8 pounds, we will apply the following steps:
1. Calculate the Z-scores for the weight boundaries, 114 and 138 pounds.
2. Use the Z-scores to find the cumulative probability for these scores.
3. Calculate the difference between the two cumulative probabilities to obtain the proportion of runners in this weight range.
4. Convert the proportion to a percentage.
Let's go through these steps:
**Step 1: Calculate the Z-scores**
[tex]The Z-score is calculated using the formula:\[ Z = \frac{(X - \mu)}{\sigma} \]Where \( \mu \) is the mean, \( \sigma \) is the standard deviation, and \( X \) is the value in question.For \( X = 114 \) pounds (lower bound):\[ Z_{lower} = \frac{(114 - 122)}{8} \]\[ Z_{lower} = \frac{-8}{8} \]\[ Z_{lower} = -1 \]For \( X = 138 \) pounds (upper bound):\[ Z_{upper} = \frac{(138 - 122)}{8} \]\[ Z_{upper} = \frac{16}{8} \]\[ Z_{upper} = 2 \][/tex]
**Step 2: Find cumulative probabilities**
To find the cumulative probabilities for the Z-scores, we typically refer to standard normal distribution tables or use a statistical calculator. However, for the Z-scores of -1 and 2, they are common values and I can provide the probabilities based on the standard normal distribution.
[tex]For \( Z_{lower} = -1 \): The cumulative probability is approximately 0.1587.For \( Z_{upper} = 2 \): The cumulative probability is approximately 0.9772.[/tex]
**Step 3: Determine the proportion between the weights**
To find the proportion of runners between 114 pounds and 138 pounds, subtract the lower cumulative probability from the upper cumulative probability:
[tex]\[ Proportion = P(Z_{upper}) - P(Z_{lower}) \]\[ Proportion = 0.9772 - 0.1587 \]\[ Proportion = 0.8185 \][/tex]
**Step 4: Convert the proportion to a percentage**
[tex]\[ Percentage = Proportion \times 100 \]\[ Percentage = 0.8185 \times 100 \]\[ Percentage = 81.85\% \][/tex]
This means that approximately 81.85% of the runners weigh between 114 pounds and 138 pounds.
The closest answer choice is (D) 82%.
