Questions: The weights of newborn baby boys born at a local hospital are believed to have a normal distribution with a mean weight of 3245 grams and a standard deviation of 625 grams. If a newborn baby boy born at the local hospital is randomly selected, find the probability that the weight will be greater than 2620 grams. Round your answer to four decimal places.

Solution Steps

To find the probability that a newborn baby boy's weight is greater than 2620 grams, we first calculate the Z-score using the formula:

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

where:

• \( X = 2620 \) grams (the weight we are interested in),
• \( \mu = 3245 \) grams (the mean weight),
• \( \sigma = 625 \) grams (the standard deviation).

Substituting the values, we have:

\[ z = \frac{2620 - 3245}{625} = -1.0 \]

Thus, the Z-score for a weight of 2620 grams is \( z = -1.0 \).

Next, we need to find the probability that the weight is greater than 2620 grams. This can be expressed as:

\[ P(X > 2620) = P(Z > -1.0) \]

Using the properties of the standard normal distribution, we can express this as:

\[ P(Z > -1.0) = 1 - P(Z \leq -1.0) = P(Z \leq \infty) - P(Z \leq -1.0) \]

From standard normal distribution tables or cumulative distribution function (CDF) values, we know:

\[ P(Z \leq -1.0) \approx 0.1587 \]

Thus, we can calculate:

\[ P(Z > -1.0) = 1 - 0.1587 = 0.8413 \]

Final Answer