In a study to estimate the proportion of smokers among the...

In a study to estimate the proportion of smokers among the residents in a certain city and its suburbs, it is found that 24 of 104 urban residents are smokers, while 34 of 108 suburban residents are smokers. Is there a significant difference between the proportions of smokers among urban and suburban residents? Make use of a $P$-value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population based on sample data. It involves formulating a null hypothesis, which typically assumes no effect or no difference, and an alternative hypothesis that represents a deviation from the null. The test then determines whether the sample evidence is strong enough to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

In the context of comparing proportions across two groups, the null hypothesis usually states that the two population proportions are equal, while the alternative hypothesis suggests that there is a difference. This framework establishes the basis for evaluating whether any observed differences in sample proportions are statistically significant.

Two-sample Proportion Test

A two-sample proportion test is used to compare the proportions of a particular characteristic between two independent groups. It assesses whether the observed difference in sample proportions is greater than what could be expected due to random variation. This test often employs a z-test when the sample sizes are sufficiently large, enabling the use of normal approximation for the sampling distribution.

P-value

The p-value represents the probability of obtaining an effect at least as extreme as the one observed in the sample data, assuming that the null hypothesis is true. In the context of comparing proportions, a small p-value suggests that the observed difference between groups is unlikely to have occurred by chance under the null hypothesis, thus supporting the alternative hypothesis.

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