Question

High blood pressure has been identified as a risk factor for heart attacks and strokes. The proportion of U.S. adults with high blood pressure is 0.2. A sample of 37 U.S. adults is chosen. Use the TI-84 Plus Calculator as needed. Round the answer to at least four decimal places.
Part 1
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? It is not_______ appropriate to use the normal curve, since np = 7.4 ______< 10 and n (1 – p) = 29.6 2 10.
Part 2
A new sample of 82 adults is drawn. Find the probability that more than 32% of the people in this sample have high blood pressure. The probability that more than 32% of the people in this sample have high blood pressure is__________ X 5

Answer 1

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:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

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 $p = 0.2, n = 37$

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 $\mu = p = 0.2$

Standard deviation $s = \sqrt{\frac{0.2*0.8}{82}} = 0.0442$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.32 - 0.2}{0.0442]$

$Z = 2.72$

$Z = 2.72$ 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%.