Question

According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year’s rate. Nine out of 70 randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still 14% or if it has decreased.

Short answer

Confidence Interval: (0.0502, 0.2070)

Step-by-step solution

Find H0 and Ha: We want to test if New York City (NYC) adults take less than 14% of them to smoke, on average.

$H_0:p=0.14;H_a:p<0.14$

Determine the distribution needed: In words, CLEARLY state what your random variable P′ represents.Let P′ = the proportion of NYC residents that smoke.

Normal:$N\left(0.14,\sqrt{\frac{(0.14)(1-0.14)}{70}}\right)$

Test Statistic:$z=–0.2756$

Calculate the p-value using the normal distribution for proportions:

$p-value=0.3914$

In one to two complete sentences, explain what the p-value means for this problem.

If the null hypothesis is true (the proportion is 0.14), then there is a 0.3914 probability that the sample (estimated) proportion is 0.129 or more.

Compare α and the p-value:Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.

alpha	decision	reason for decision
0.05	Do not reject the null hypothesis.	$p-value>0.05$

Conclusion: At the 5% significance level, there is insufficient evidence to conclude that the proportion of NYC residents who smoke is less than 0.14.

Confidence Interval

Confidence Interval: (0.0502, 0.2070): The “plus-4s” confidence interval (see chapter 8) is (0.0676, 0.2297).