Question

National data in the 1960's showed that about 44% of the adult population had never smoked cigarettes. In 1995 a national health survey interviewed a random sample of 881 adults and found that 52% had never been smokers.
a) Create a 95% confidence interval for the proportion of adults (in 1995) who had never been smokers.
b) Does this provide evidence of a change in behavior among Americans? Using your confidence level, test an appropriate hypothesis and state your conclusion.

Answer 1

Answer:

a) $0.52 - 1.96\sqrt{\frac{0.52(1-0.52)}{881}}=0.487$

$0.52 + 1.96\sqrt{\frac{0.52(1-0.52)}{881}}=0.553$

The 95% confidence interval would be given by (0.487;0.553)

b) After see the confidence interval we see that the lower limits 0.487>0.44 with 0.44 or 44% the estimated  proportion for 1960, we can conclude that we have a significant increase in the adult proportion had never smoked cigarettes.

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Solution to the problem

Part a

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by $\alpha=1-0.95=0.05$ and $\alpha/2 =0.025$. And the critical value would be given by:

$z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96$

The confidence interval for the mean is given by the following formula:  

$\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}$

If we replace the values obtained we got:

$0.52 - 1.96\sqrt{\frac{0.52(1-0.52)}{881}}=0.487$

$0.52 + 1.96\sqrt{\frac{0.52(1-0.52)}{881}}=0.553$

The 95% confidence interval would be given by (0.487;0.553)

Part b

After see the confidence interval we see that the lower limits 0.487>0.44 with 0.44 or 44% the estimated  proportion for 1960, we can conclude that we have a significant increase in the adult proportion had never smoked cigarettes at 5% of significance.