Respuesta :
Using the normal distribution, we have that:
a. 68% of pregnancies last between 250 and 282 days.
b. 99.7% of all pregnancies fall between 218 days and 314 days.
c. A pregnancy located in the 16th percentile would last 250 days.
d. There is a 0.2140 = 21.40% probability that 5 of them have a pregnancy that last between 250 and 266 days.
e. Her pregnancy length will be of at least 286 days.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
The mean and the standard deviation are given, respectively, by:
[tex]\mu = 266, \sigma = 16[/tex]
For item a, the proportion is the p-value of Z when X = 282 subtracted by the p-value of Z when X = 250, hence:
X = 282:
Z = (282 - 266)/16
Z = 1 has a p-value of 0.84.
X = 250:
Z = (250 - 266)/16
Z = -1 has a p-value of 0.16.
0.84 - 0.16 = 0.68.
68% of pregnancies last between 250 and 282 days.
For item b, By the empirical rule, 99.7% of the measures of a normal distribution fall within 3 standard deviations of the mean, hence:
- 266 - 3 x 16 = 218 days.
- 266 + 3 x 16 = 314 days.
99.7% of all pregnancies fall between 218 days and 314 days.
For item c, the pregnancy length is X when Z = -1, hence:
-1 = (X - 266)/16
X - 266 = -16
X = 250.
A pregnancy located in the 16th percentile would last 250 days.
For item d, using the binomial distribution with p = 0.34(due to item a and the symmetry of the normal distribution) and n = 15, the probability is given as follows:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{15,5}.(0.34)^{5}.(0.66)^{10} = 0.2140[/tex]
There is a 0.2140 = 21.40% probability that 5 of them have a pregnancy that last between 250 and 266 days.
For item e, her pregnancy length is at the 90th percentile, that is, X when Z = 1.28, hence:
1.28 = (X - 266)/16
X - 266 = 1.28 x 16
X = 286.
Her pregnancy length will be of at least 286 days.
More can be learned about the normal distribution at https://brainly.com/question/4079902
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