Questions: The weight of adult male grizzly bears living in the wild in the continental United States is approximately normally distributed with a mean of 500 pounds and a standard deviation of 50 pounds. The weight of adult female grizzly bears is approximately normally distributed with a mean of 300 pounds and a standard deviation of 40 pounds. Approximately, what would be the weight of a female grizzly bear with the same standardized score (z-score) as a male grizzly bear with a weight of 530 pounds? (A) 276 pounds (B) 324 pounds (C) 330 pounds (D) 340 pounds (E) 530 pounds

Solution Steps

To find the z-score for a male grizzly bear weighing 530 pounds, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

• \( X = 530 \) (weight of the male grizzly bear),
• \( \mu = 500 \) (mean weight of adult male grizzly bears),
• \( \sigma = 50 \) (standard deviation of the weight of adult male grizzly bears).

Substituting the values, we have:

\[ z = \frac{530 - 500}{50} = \frac{30}{50} = 0.6 \]

Thus, the z-score for the male grizzly bear weighing 530 pounds is \( z = 0.6 \).

Next, we use the same z-score to find the weight of a female grizzly bear. The formula to find the value (weight) based on the z-score is rearranged as follows:

\[ X = z \cdot \sigma + \mu \]

For the female grizzly bear:

• \( \mu = 300 \) (mean weight of adult female grizzly bears),
• \( \sigma = 40 \) (standard deviation of the weight of adult female grizzly bears).

Substituting the values, we have:

\[ X = 0.6 \cdot 40 + 300 = 24 + 300 = 324 \]

Thus, the weight of the female grizzly bear with the same z-score is \( 324 \) pounds.

Final Answer