Suppose that the lengths of human pregnancies are normally distributed with a mean of 266 days and a standard deviation of 16 days. Complete the following statements.
(a) Approximately ? of pregnancies have lengths between 234 days and 298 days.
(b) Approximately 68% of pregnancies have lengths between ? days and ? days.
Answer
So, approximately 68% of pregnancies have lengths between \(\boxed{250}\) days and \(\boxed{282}\) days.
Step 1 ：Given that the lengths of human pregnancies are normally distributed with a mean of 266 days and a standard deviation of 16 days.
Step 2 ：For part (a), we need to calculate the z-scores for 234 days and 298 days. The z-score is calculated as \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：Calculating the z-scores, we get \(z_{234} = \frac{234 - 266}{16} = -2.0\) and \(z_{298} = \frac{298 - 266}{16} = 2.0\).
Step 4 ：We then use the standard normal distribution to find the proportion of pregnancies that fall within this range. The standard normal distribution tells us that approximately 95.45% of data falls within 2 standard deviations of the mean.
Step 5 ：So, approximately \(\boxed{95.45\%}\) of pregnancies have lengths between 234 days and 298 days.
Step 6 ：For part (b), we know that approximately 68% of data falls within one standard deviation of the mean in a normal distribution. So we just need to calculate the range of days that corresponds to one standard deviation above and below the mean.
Step 7 ：Calculating this, we get \(\mu - \sigma = 266 - 16 = 250\) days and \(\mu + \sigma = 266 + 16 = 282\) days.
Step 8 ：So, approximately 68% of pregnancies have lengths between \(\boxed{250}\) days and \(\boxed{282}\) days.
