Questions: Provide an appropriate response. Use the Standard Normal Table to find the probability. The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard deviation of 15 days. A baby is premature if it is born three weeks early. What percent of babies are born prematurely?

Solution Steps

To determine the probability of a baby being born prematurely, we first calculate the Z-score for the threshold of premature birth, which is defined as being born 21 days early. The threshold is given by:

\[ X = 268 - 21 = 247 \text{ days} \]

The Z-score is calculated using the formula:

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

Substituting the values:

\[ z = \frac{247 - 268}{15} = \frac{-21}{15} = -1.4 \]

Thus, the Z-score for the premature threshold is:

\[ Z = -1.4 \]

Next, we find the probability that a baby is born prematurely, which corresponds to the area under the normal distribution curve to the left of the Z-score we calculated. This is represented mathematically as:

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

Since \(\Phi(-\infty) = 0\), we have:

\[ P = \Phi(-1.4) \]

Using the standard normal distribution table, we find:

\[ \Phi(-1.4) \approx 0.0808 \]

Thus, the probability of a baby being born prematurely is:

\[ P \approx 0.0808 \]

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

\[ \text{Percentage} = 0.0808 \times 100 \approx 8.08\% \]

Final Answer