Answer:
1. It is not appropriate to use the normal curve, since np = 7.4 < 10.
2. The probability that more than 32% of the people in this sample have high blood pressure is 0.0033 = 0.33%.
Step-by-step explanation:
Binomial approximation to the normal:
The binomial approximation to the normal can be used if:
np >= 10 and n(1-p) >= 10
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
![E(X) = np](https://tex.z-dn.net/?f=E%28X%29%20%3D%20np)
The standard deviation of the binomial distribution is:
![\sqrt{V(X)} = \sqrt{np(1-p)}](https://tex.z-dn.net/?f=%5Csqrt%7BV%28X%29%7D%20%3D%20%5Csqrt%7Bnp%281-p%29%7D)
Normal probability distribution
Problems of normally distributed distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
and standard deviation ![s = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
The proportion of U.S. adults with high blood pressure is 0.2. A sample of 37 U.S. adults is chosen.
This means, respectively, that ![p = 0.2, n = 37](https://tex.z-dn.net/?f=p%20%3D%200.2%2C%20n%20%3D%2037)
Is it appropriate to use the normal approximation to find the probability that more than 48% of the people in the sample have high blood pressure?
np = 37*0.2 = 7.4 < 10
So not appropriate.
It is not appropriate to use the normal curve, since np = 7.4 < 10.
Part 2:
Now n = 82, 82*0.2 = 16.4 > 10, so ok
Mean and standard deviation:
By the Central Limit Theorem,
Mean ![\mu = p = 0.2](https://tex.z-dn.net/?f=%5Cmu%20%3D%20p%20%3D%200.2)
Standard deviation ![s = \sqrt{\frac{0.2*0.8}{82}} = 0.0442](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7B0.2%2A0.8%7D%7B82%7D%7D%20%3D%200.0442)
Find the probability that more than 32% of the people in this sample have high blood pressure.
This probability is 1 subtracted by the pvalue of Z when X = 0.32. 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{0.32 - 0.2}{0.0442]](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B0.32%20-%200.2%7D%7B0.0442%5D)
![Z = 2.72](https://tex.z-dn.net/?f=Z%20%3D%202.72)
has a pvalue of 0.9967.
1 - 0.9967 = 0.0033
The probability that more than 32% of the people in this sample have high blood pressure is 0.0033 = 0.33%.