Question

The data from National Statistics Office (NSO) for 2008 showed that about 44% of the adult population had never smoked.

a. State a null and alternative hypothesis to test that the fraction of the 2018 population of adults that had never smoked had increased
b. A national random sample of 891 adults were interviewed and 463 stated that they had never smoked. Perform a z-test of the hypothesis and give an appropriate p-value
c. Create a 98% confidence interval for the proportion of adults who had never been smokers
d. Give the value of the power function π (p) for p=0.46, 0.48, 0.50, 0.52 with the choice of α=0.02 and a "greater than" alternative hypothesis.
e. Compute the power function for these values if we increase the sample to 1600. Explain why these values increased

Answer 1

The hypothesis test aims to determine if the proportion of adults who have never smoked increased from 44%. A z-test is performed based on sample data, and a confidence interval is created to estimate the true proportion. Calculating the power function with different assumed true proportions and sample sizes illustrate how power increases with larger samples.

Step-by-step explanation:

Null and Alternative Hypotheses

To test if the fraction of the 2018 adult population that had never smoked increased compared to the 2008 data provided by the National Statistics Office (NSO), we formulate the following hypotheses:

• Null Hypothesis (H0): p = 0.44 - The proportion of adults in 2018 who have never smoked is equal to 44%.
  
• Alternative Hypothesis (Ha): p > 0.44 - The proportion of adults in 2018 who have never smoked is greater than 44%.
  

Z-Test Calculation

From the sample of 891 adults, 463 stated that they had never smoked, giving us a sample proportion (p') of 0.52. To perform a z-test:

1. Calculate the standard error (SE) of the sample proportion: SE = sqrt[(0.44*(1-0.44))/891].
2. Compute the z-score: z = (p' - 0.44)/SE.
3. Compare the z-score to the standard normal distribution to find the p-value.

98% Confidence Interval

We use the z-score corresponding to a 98% confidence level and the sample proportion to determine the confidence interval for the proportion of adults who have never been smokers.

Power Function Calculation

To give the value of the power function π (p) for p = 0.46, 0.48, 0.50, and 0.52, we calculate the probability of correctly rejecting the null hypothesis for these assumed true proportions.

Effect of Increased Sample Size on Power

Increasing the sample size to 1600 increases the power function values. This is because a larger sample size reduces the standard error of the estimate, making it easier to detect a difference if one truly exists, thereby increasing the power of the test.