According to an article in Bloomberg Business week, New York...

According to an article in Bloomberg Business week, 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.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. In the given problem, the goal is to evaluate whether the current smoking rate in New York City has decreased from a known benchmark of 14%, which involves testing a null hypothesis against an alternative hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the status quo or a specified value (here, p = 0.14), while the alternative hypothesis (H1) reflects the claim being tested (in this case, that the true proportion is less than 0.14). This sets up a one-tailed test aimed at detecting a decrease in the smoking rate.

One-Sample Proportion Test

A one-sample proportion test is used when comparing an observed sample proportion to a known or hypothesized population proportion. The test assesses whether the observed deviation from the benchmark is statistically significant, accounting for the variability inherent in sampling.

Z-Test for Proportions

The Z-test for proportions is applied because under certain conditions (such as a sufficiently large sample size), the sampling distribution of the sample proportion can be approximated by a normal distribution. The test statistic is calculated by subtracting the hypothesized proportion from the observed proportion and dividing by the standard error of the proportion.

Standard Error

The standard error in this context quantifies the expected variability of the sample proportion under the null hypothesis. It is calculated as the square root of [p(1-p)/n], where p is the hypothesized proportion and n is the sample size, determining the spread of the sampling distribution.

P-Value and Significance Level

The p-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed result, assuming the null hypothesis is true. Comparing the p-value to a predetermined significance level (commonly 0.05) helps to decide whether to reject the null hypothesis and conclude that the smoking rate has decreased.

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