Questions: The weights for 12-month-old males are normally distributed with a mean of 22.6 pounds and a standard deviation of 3.0 pounds. Use the given table to find the percentage of 12-month-old males who weigh between 16 and 18.4 pounds. 7% of 12-month-old males weigh between 16 and 18.4 pounds.

Solution Steps

To find the percentage of 12-month-old males who weigh between 16 and 18.4 pounds, we first calculate the Z-scores for the given weights using the formula:

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

where:

• \(X\) is the value,
• \(\mu = 22.6\) pounds (mean),
• \(\sigma = 3.0\) pounds (standard deviation).

Calculating the Z-scores:

• For \(X = 16\): \[ Z_{start} = \frac{16 - 22.6}{3.0} = -2.2 \]
• For \(X = 18.4\): \[ Z_{end} = \frac{18.4 - 22.6}{3.0} = -1.4 \]

Next, we find the probability that a 12-month-old male weighs between 16 and 18.4 pounds using the cumulative distribution function \( \Phi \):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.4) - \Phi(-2.2) \]

From the calculations, we find: \[ P = 0.0669 \]

To express the probability as a percentage, we multiply by 100:

\[ \text{Percentage} = P \times 100 = 0.0669 \times 100 = 6.69\% \]

Final Answer