The length of human pregnancies from conception to birth varies according to a distribution that can be modeled by a normal random variable with mean 265 days and standard deviation 14 days. Question 1. What percent of pregnancies last less than 240 days? Note that the answer is requested as a percent. Use 2 decimal places in your answer. % Question 2. What percent of pregnancies last between 240 and 270 days? Note that the answer is requested as a percent. Use 2 decimal places in your answer. % Question 3. The longest 20% of pregnancies last at least how many days? (round to the nearest whole day)

1) 3.67%

2) 60.39%

3) The longest 20% of pregnancies last at least 277 days.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Question 1. What percent of pregnancies last less than 240 days?

This is the pvalue of Z when X = 240. So

has a pvalue of 0.0367

3.67% of pregnancies last less than 240 days

Question 2. What percent of pregnancies last between 240 and 270 days?

This is the pvalue of Z when X = 270 subtracted by the pvalue of Z when X = 240. So

X = 270

has a pvalue of 0.6406

X = 240

has a pvalue of 0.0367

0.6406 - 0.0367 = 0.6039

60.39% of pregnancies last between 240 and 270 days

3. The longest 20% of pregnancies last at least how many days?

They last at least X days, in which X is found when Z has a pvalue of 1-0.2 = 0.8. So it is X when Z = 0.84.

The longest 20% of pregnancies last at least 277 days.

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