Question

The National Health Interview Survey conducted of 33,326 adults by the U.S. National Center for Health Statistics in 2003 indicated that 21.4% of adults were current smokers. A similar study conducted in 1991 of 42,000 adults indicated that 25.6% were current smokers. We will use the subscript "1" for the year 1991 and the subscript "2" for the year 2003.
a. Find and interpret a point estimate of the differencebetween the proportion of current smokers in 1991 and theproportion of current smokers in 2003.
b. A 99% confidence interval for the true difference is(0.032, 0.050). Interpret.
c. What assumptions must you make for the interval in (b) tobe valid?

Answer 1

Answer:

Step-by-step explanation:

Hello!

You have the data of two The National Health Interview Survey conducted in 1991 and 2003 by the U.S. National Center for Health Statistics.

Let

X₁: The number of adults that were smokers in 1991

n₁= 42000

p'₁= 0.256

X₂: The number of adults that were smokers in 2003

n₂= 33326

p'₂= 0.214

The parameter of interest is the difference between the population proportion of adult smokers in 1991 and the population proportion of adult smokers in 2003, symbolically: "p₁ - p₂"

a) The point estimate of "p₁ - p₂" ios the difference between the sample proportions:

p'₁ - p'₂= 0.256-0.214= 0.042

b) and c)

To estimate the difference of proportions using a CI, you have to apply the central limit theorem and approximate the distribution of the sample proportions to normal. for this approximation to be valid, the sample size for both populations should be high enough (n≥30) and the populations should be at least 20 times bigger than the sample taken. Both samples should be random and independent. Each sample must have at least 10 successes and at least 10 failures. If all conditions are met you can calculate the 99% CI as:

(p'₁-p'₂)±$Z_{1-\alpha /2}$ * $\sqrt{\frac{p'_1(1-p'_1){n_1} +\frac{p'_2(1-p'_2)}{n_2} }$

$Z_{1-\alpha /2}= Z_{0.995}= 2.586$

(0.042) ± 2.586* $\sqrt{\frac{0.256*0.744}{42000} +\frac{0.214*0.786}{33326} }$

[0.034; 0.050]

Using a 99% confidence level you'd expect that the interval [0.034; 0.050] will include the true value of the difference between the proportion of adult smokers in 1991 and the proportion of adult smokers in 2003.

I hope this helps!