Respuesta :
Answer:
a) 281 days.
b) 255 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 = 8[/tex]
(a) What is the minimum pregnancy length that can be in the top 8% of pregnancy lengths?
100 - 8 = 92th percentile.
X when Z has a pvalue of 0.92. So X when Z = 1.405.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.405 = \frac{X - 270}{8}[/tex]
[tex]X - 270 = 1.405*8[/tex]
[tex]X = 281[/tex]
(b) What is the maximum pregnancy length that can be in the bottom 3% of pregnancy lengths?
3rd percentile.
X when Z has a pvalue of 0.03. So X when Z = -1.88
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.88 = \frac{X - 270}{8}[/tex]
[tex]X - 270 = -1.88*8[/tex]
[tex]X = 255[/tex]
