Respuesta :
Answer:
1) The most typical 68% of pregnancies last between 250 and 282 days, the most typical 95% between 234 and 298 days, and the most typical 99.7% between 218 and 314 days.
2) 15.87% of all pregnancies last less than 250 days
2a) 83.5% of pregnancies last between 241 and 286 days
2b) 10.56% of pregnancies last more than 286 days.
2c) 0% of pregnancies last more than 333 days
3) A pregnancy length of 234.6 days cuts off the shortest 2.5% of pregnancies.
4) The first quartile of pregnancy lengths is of 255.2, and the third quartile is of 276.8 days.
5) The most typical 72% of all pregnancies last between 248.72 and 283.28 days.
Step-by-step explanation:
Empirical Rule:
The Empirical Rule states that, for a normally distributed random variable:
Approximately 68% of the measures are within 1 standard deviation of the mean.
Approximately 95% of the measures are within 2 standard deviations of the mean.
Approximately 99.7% of the measures are within 3 standard deviations of the mean.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Mean of 266 days and a standard deviation of 16 days.
This means that [tex]\mu = 266, \sigma = 16[/tex]
(1) Using the 68-95-99.7% rule, between what two lengths do the most typical 68% of all pregnancies fall 95%, 99.7%?
68%: within 1 standard deviation of the mean, so 266 - 16 = 250 days to 266 + 16 = 282 days.
95%: within 2 standard deviations of the mean, so 266 - 32 = 234 days to 266 + 32 = 298 days.
99.7%: within 3 standard deviations of the mean, so 266 - 48 = 218 days to 266 + 48 = 314 days.
The most typical 68% of pregnancies last between 250 and 282 days, the most typical 95% between 234 and 298 days, and the most typical 99.7% between 218 and 314 days.
(2) What percent of all pregnancies last less than 250 days?
The proportion is the p-value of Z when X = 250. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{250 - 266}{16}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a p-value of 0.1587.
0.1587*100% = 15.87%.
15.87% of all pregnancies last less than 250 days.
(a) What percentage of pregnancies last between 241 and 286 days?
The proportion is the p-value of Z when X = 286 subtracted by the p-value of Z when X = 241. So
X = 286
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{286 - 266}{16}[/tex]
[tex]Z = 1.25[/tex]
[tex]Z = 1.25[/tex] has a p-value of 0.8944.
X = 241
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{241 - 266}{16}[/tex]
[tex]Z = -1.56[/tex]
[tex]Z = -1.56[/tex] has a p-value of 0.0594.
0.8944 - 0.0594 = 0.835*100% = 83.5%
83.5% of pregnancies last between 241 and 286 days.
(b) What percentage of pregnancies last more than 286 days?
1 - 0.8944 = 0.1056*100% = 10.56%.
10.56% of pregnancies last more than 286 days.
(c) What percentage of pregnancies last more than 333 days?
The proportion is 1 subtracted by the p-value of Z = 333. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{333 - 266}{16}[/tex]
[tex]Z = 4.19[/tex]
[tex]Z = 4.19[/tex] has a p-value of 1
1 - 1 = 0% of pregnancies last more than 333 days.
(3) What length cuts off the shortest 2.5% of pregnancies?
This is the 2.5th percentile, which is X when Z = -1.96.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.96 = \frac{X - 266}{16}[/tex]
[tex]X - 266 = -1.96*16[/tex]
[tex]X = 234.6[/tex]
A pregnancy length of 234.6 days cuts off the shortest 2.5% of pregnancies.
(4) Find the quartiles for pregnancy length.
First quartile the 25th percentile, which is X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 266}{16}[/tex]
[tex]X - 266 = -0.675*16[/tex]
[tex]X = 255.2[/tex]
Third quartile is the 75th percentile, so X when Z = 0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.675 = \frac{X - 266}{16}[/tex]
[tex]X - 266 = 0.675*16[/tex]
[tex]X = 276.8[/tex]
The first quartile of pregnancy lengths is of 255.2, and the third quartile is of 276.8 days.
(5) Between what two lengths are the most typical 72% of all pregnancies?
Between the 50 - (72/2) = 14th percentile and the 50 + (72/2) = 86th percentile.
14th percentile:
X when Z = -1.08.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.08 = \frac{X - 266}{16}[/tex]
[tex]X - 266 = -1.08*16[/tex]
[tex]X = 248.72[/tex]
86th percentile:
X when Z = 1.08.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.08 = \frac{X - 266}{16}[/tex]
[tex]X - 266 = 1.08*16[/tex]
[tex]X = 283.28[/tex]
The most typical 72% of all pregnancies last between 248.72 and 283.28 days.
