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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:
The standard deviation of the binomial distribution is:
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:
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
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
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
Standard deviation
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
By the Central Limit Theorem
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%.