Over the past few decades, public health officials have...

Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? After conducting the test, your decision and conclusion are a. Reject $H_{0}:$ There is sufficient evidence to conclude that more than $30 \%$ of teen girls smoke to stay thin. b. Do not reject $H_{0}:$ There is not sufficient evidence to conclude that less than $30 \%$ of teen girls smoke to stay thin. c. Do not reject $H_{0}:$ There is not sufficient evidence to conclude that more than $30 \%$ of teen girls smoke to stay thin. d. Reject $H_{0}:$ There is sufficient evidence to conclude that less than $30 \%$ of teen girls smoke to stay thin.

Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? After conducting the test, your decision and conclusion are
a. Reject $H_{0}:$ There is sufficient evidence to conclude that more than $30 \%$ of teen girls smoke to stay thin.
b. Do not reject $H_{0}:$ There is not sufficient evidence to conclude that less than $30 \%$ of teen girls smoke to stay thin.
c. Do not reject $H_{0}:$ There is not sufficient evidence to conclude that more than $30 \%$ of teen girls smoke to stay thin.
d. Reject $H_{0}:$ There is sufficient evidence to conclude that less than $30 \%$ of teen girls smoke to stay thin.

Key Concepts

Test Statistic and P-Value

The test statistic is calculated from sample data and reflects how far the observed sample proportion deviates from what is expected under the null hypothesis. The p-value quantifies the probability of observing a test statistic at least as extreme as the one calculated, given that the null hypothesis is true, and is used to make a decision about rejecting or not rejecting H?.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves setting up competing hypotheses (null and alternative) and using a test statistic to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis, typically denoted as H?, represents the default or status quo claim about a population parameter, while the alternative hypothesis, denoted as H?, reflects what the researcher aims to support. In tests regarding proportions, H? might assert that the true proportion equals a specified value, and H? would claim that the proportion is either greater than, less than, or not equal to that value.

Sample Proportion

The sample proportion is the ratio of the number of successes (events of interest) to the total sample size. It serves as an estimate of the corresponding population proportion and is a key statistic in tests of proportions, providing the basis for calculating the test statistic.

Significance Level and Decision Rule

The significance level, often denoted by ?, is a threshold for determining whether the p-value is sufficiently small to reject the null hypothesis. The decision rule is based on this comparison: if the p-value is less than or equal to ?, the null hypothesis is rejected; otherwise, the null hypothesis is not rejected. This rule helps in controlling the probability of making a Type I error.

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