Answer:
We have a normal distribution with a mean of 267 days and a standard deviation of 15 days. To solve this proble we're going to need the help of a calculator.
a. The probability of a pregnancy lasting 309 days or longer is:
P(z>309) = 0.0026 or 0.26%
b. The lowest 4% is separeted by the 240.74 days. The probability of pregnancy lasting 240.74 days is 4%.
Answer:
a) 0.26% probability of a pregnancy lasting 309 days or longer.
b) A pregnancy length of 241 days separates premature babies from those who are not premature.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 267, \sigma = 15[/tex]
a. Find the probability of a pregnancy lasting 309 days or longer.
This is 1 subtracted by the pvalue of Z when X = 309. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{309 - 267}{15}[/tex]
[tex]Z = 2.8[/tex]
[tex]Z = 2.8[/tex] has a pvalue of 0.9974
So there is a 1-0.9974 = 0.0026 = 0.26% probability of a pregnancy lasting 309 days or longer.
b. If the length of pregnancy is in the lowest 4%, then the baby is premature. Find the length that separates premature babies from those who are not premature.
This is the value of X when Z has a pvalue of 0.04. So X when Z = -1.75
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.75 = \frac{X - 267}{15}[/tex]
[tex]X - 267 = -1.75*15[/tex]
[tex]X = 240.75[/tex]
A pregnancy length of 241 days separates premature babies from those who are not premature.
