Answer:
0.1003 = 10.03% probability that a simple random sample of size n= 10 results in a sample mean greater than 40 inches.
Gestation periods:
1) 0.3539 = 35.39% probability a randomly selected pregnancy lasts less than 260 days.
2) 0.0465 = 4.65% probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less.
3) 0.004 = 0.4% probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less.
4) 0.9844 = 98.44% probability a random sample of size 15 will have a mean gestation period within 10 days of the mean.
Step-by-step explanation:
To solve these questions, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The height, X, of all 3-year-old females is approximately normally distributed with mean 38.72 inches and standard deviation 3.17 inches.
This means that ![\mu = 38.72, \sigma = 3.17](https://tex.z-dn.net/?f=%5Cmu%20%3D%2038.72%2C%20%5Csigma%20%3D%203.17)
Sample of 10:
This means that ![n = 10, s = \frac{3.17}{\sqrt{10}}](https://tex.z-dn.net/?f=n%20%3D%2010%2C%20s%20%3D%20%5Cfrac%7B3.17%7D%7B%5Csqrt%7B10%7D%7D)
Compute the probability that a simple random sample of size n= 10 results in a sample mean greater than 40 inches.
This is 1 subtracted by the p-value of Z when X = 40. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{40 - 38.72}{\frac{3.17}{\sqrt{10}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B40%20-%2038.72%7D%7B%5Cfrac%7B3.17%7D%7B%5Csqrt%7B10%7D%7D%7D)
![Z = 1.28](https://tex.z-dn.net/?f=Z%20%3D%201.28)
has a p-value of 0.8997
1 - 0.8997 = 0.1003
0.1003 = 10.03% probability that a simple random sample of size n= 10 results in a sample mean greater than 40 inches.
Gestation periods:
![\mu = 266, \sigma = 16](https://tex.z-dn.net/?f=%5Cmu%20%3D%20266%2C%20%5Csigma%20%3D%2016)
1. What is the probability a randomly selected pregnancy lasts less than 260 days?
This is the p-value of Z when X = 260. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{260 - 266}{16}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B260%20-%20%20266%7D%7B16%7D)
![Z = -0.375](https://tex.z-dn.net/?f=Z%20%3D%20-0.375)
has a p-value of 0.3539.
0.3539 = 35.39% probability a randomly selected pregnancy lasts less than 260 days.
2. What is the probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less?
Now
, so:
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{260 - 266}{\frac{16}{\sqrt{20}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B260%20-%20266%7D%7B%5Cfrac%7B16%7D%7B%5Csqrt%7B20%7D%7D%7D)
![Z = -1.68](https://tex.z-dn.net/?f=Z%20%3D%20-1.68)
has a p-value of 0.0465.
0.0465 = 4.65% probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less.
3. What is the probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less?
Now
, so:
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{260 - 266}{\frac{16}{\sqrt{50}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B260%20-%20266%7D%7B%5Cfrac%7B16%7D%7B%5Csqrt%7B50%7D%7D%7D)
![Z = -2.65](https://tex.z-dn.net/?f=Z%20%3D%20-2.65)
has a p-value of 0.0040.
0.004 = 0.4% probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less.
4. What is the probability a random sample of size 15 will have a mean gestation period within 10 days of the mean?
Sample of size 15 means that
. This probability is the p-value of Z when X = 276 subtracted by the p-value of Z when X = 256.
X = 276
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{276 - 266}{\frac{16}{\sqrt{15}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B276%20-%20266%7D%7B%5Cfrac%7B16%7D%7B%5Csqrt%7B15%7D%7D%7D)
![Z = 2.42](https://tex.z-dn.net/?f=Z%20%3D%202.42)
has a p-value of 0.9922.
X = 256
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{256 - 266}{\frac{16}{\sqrt{15}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B256%20-%20266%7D%7B%5Cfrac%7B16%7D%7B%5Csqrt%7B15%7D%7D%7D)
![Z = -2.42](https://tex.z-dn.net/?f=Z%20%3D%20-2.42)
has a p-value of 0.0078.
0.9922 - 0.0078 = 0.9844
0.9844 = 98.44% probability a random sample of size 15 will have a mean gestation period within 10 days of the mean.