Question

8. The lengths of pregnancies are normally distributed with a mean of 267 days and a standard deviation of 15 days. (a) Find the probability that an individual woman has a pregnancy shorter than 259 days. (b) If 36 women are randomly selected, find the probability that they have a mean preg- nancy shorter than 259 days. (c) There should be a difference in your method for the previous two questions. Explain what you did differently for each problem and explain WHY your answers are different. (d) Find the probability that an individual woman has a pregnancy longer than 295 days. (e) Find the probability that an individual woman has a pregnancy between 259 and 295 days. (f) What is the cutoff number of days for a pregnancy for the top 15% of women? (g) What is the cutoff number of days for a pregnancy for the bottom 25% of women?

please answer it all not using excel

Answer 1

(a) Let random variable X mean the number of days of pregnancy. We will convert it to standard normal variable Z. P(X < 259) = P(Z < (259- 267)/15) = P(Z < -0.5333) = 0.29691

(b) Let mean pregnancy for 36 women be .

X

Note that if X is distributed normally with mean $\mu$ and standard deviation $\sigma$ , then
X
is distributed normally with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$ .

Hence, $P(\bar X < 259)$ = P(Z < (259 - 267)/(15/6)) = P(Z < -3.2) = 0.00068714

(c) The difference is that for (a) and (b) the random variables are different. In (a) we have the normally distributed pregnancy days as the random variable and in (b) we have the sample mean of pregnancy days as the random variable. The sample mean of a normally distributed random variable is also distribued normally but with a dfferent standard deviation (though same mean). Hence the answers of (b) and (c) are different as evident by the calculations.

(d) P(X > 295) = P(Z > (295 - 267)/15) = P(Z > 1.867) = 1 - P(Z < 1.867) = 0.030951

(e) P(259 < X < 295) = P((259 - 267)/15 < Z < (295 - 267)/15) = P(-0.53333 < Z < 1.86667)

=P(Z < 1.86667) - P(Z < -0.53333) = 0.96905 - 0.2969 = 0.672

(f) We need to find k such that P(X > k) = 0.15

Or P(Z > (k -267)/15) = 0.15.

1 - P(Z < (k - 267)/15) = 0.15

P(Z < (k - 267)/15) = 0.85

(k - 267)/15 = 1.0364

k = 282.55

(g) We need to find m such that P(X < m) = 0.25

Or P(Z < (m - 267)/15) = 0.25

(m - 267)/15 = 0.6745

m = 277.12