Questions: The length of human pregnancies is approximately normal with mean μ=266 days and standard deviation σ=16 days. Complete parts (a) through (f). (a) What is the probability that a randomly selected pregnancy lasts less than 260 days? The probability that a randomly selected pregnancy lasts less than 260 days is approximately (Round to four decimal places as needed.)

Solution Steps

To find the probability that a randomly selected pregnancy lasts less than 260 days, we first calculate the Z-score using the formula:

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

where:

• \(X = 260\) (the value we are interested in),
• \(\mu = 266\) (the mean of the distribution),
• \(\sigma = 16\) (the standard deviation of the distribution).

Substituting the values, we have:

\[ z = \frac{260 - 266}{16} = \frac{-6}{16} = -0.375 \]

Thus, the Z-score for the value 260 is:

\[ z = -0.375 \]

Next, we calculate the probability that a randomly selected pregnancy lasts less than 260 days. This is given by the cumulative distribution function (CDF) of the normal distribution:

\[ P(X < 260) = \Phi(Z_{end}) - \Phi(Z_{start}) \]

In this case, \(Z_{end} = -0.375\) and \(Z_{start} = -\infty\). Therefore, we have:

\[ P(X < 260) = \Phi(-0.375) - \Phi(-\infty) \]

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

\[ P(X < 260) = \Phi(-0.375) \]

Using the standard normal distribution table or calculator, we find:

\[ \Phi(-0.375) \approx 0.3538 \]

Final Answer