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