Respuesta :
Answer:
a) The minimum pregnancy length is 27.1 days.
b) The maximum pregnancy length is 265.2 days.
Step-by-step explanation:
Problems of normally distributed samples are 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 = 270, \sigma = 11[/tex]
(a) What is the minimum pregnancy length that can be in the top 88% of pregnancy lengths?
This is the value of X when Z has a pvalue of 1-0.88 = 0.12. So it is X when Z = -1.175.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.175 = \frac{X - 270}{11}[/tex]
[tex]X - 270 = -1.175*11[/tex]
[tex]X = 257.1[/tex]
The minimum pregnancy length is 27.1 days.
(b) What is the maximum pregnancy length that can be in the bottom 33% of pregnancy lengths?
This is the value of Z when Z has a pvalue of 0.33. So it is X when Z = -0.44.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.44 = \frac{X - 270}{11}[/tex]
[tex]X - 270 = -0.44*11[/tex]
[tex]X = 265.2[/tex]
The maximum pregnancy length is 265.2 days.
