Confidence Intervals and Hypothesis Tests Practice

Proportions

Teenage Smoking Habits

A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed that 15% percent of all teenagers smoke. A Gallup survey interviewed a nationally representative sample of 785 teenagers aged 13 to 17. Seventy-one (71) teenagers in the survey acknowledged having smoked at least once in the past week.

We want to explore if the study provide adequate evidence to conclude that the percentage of teenagers who smoke has changed since the original study (that is, is \(p\) now different than 15%).

QUESTION: What are the requirements for a valid hypothesis test for a proportion, and are they met?
ANSWER:

Whether or not the test requirements are met, perform the appropriate hypothesis test and create a confidence interval for the true proportion of teenagers who smoke.

Hypothesis Test

State the null and alternative hypotheses (replace the ??? with the correct information):

\[H_0: p=???\]

\[H_a: p???0.15\] Choose your significance level:

\[\alpha = 0.\]

# perform the hypothesis test:  

QUESTION: What is the value of the test statistic?
ANSWER:

QUESTION: What is the P-value?
ANSWER:

QUESTION: Based on your selected \(\alpha\) and \(P\)-value, what is your conclusion?
ANSWER:

Confidence Interval

Create a \((1-\alpha)\)% confidence interval for the true population proportion of teenagers who smoke?

Left-handed Artists

Suppose you would like to study the connection between creativity and left-handedness. Your favorite generative AI told you that the estimated population of left-handed people in the United States is about 11%. You sample 122 Visual Arts majors and find that 20 are left handed. Is this enough evidence to demonstrate that visual arts majors have a higher proportion of left handed people than the general population?

QUESTION: What is the estimated sample proportion, \(\hat{p}\)?
ANSWER:

Confidence Interval

Create a 90% confidence interval for the true population proportion of left-handed visual artists:

QUESTION: What are the requirements for a confidence interval for a proportion, and are they met?
ANSWER:

QUESTION: Interpret the confidence interval in context of the research question.
ANSWER:

Hypothesis Test

Perform a hypothesis test to determine if the proportion of left-handed Visual Art majors is higher than the general population of 11%.

QUESTION: What is the value of the test statistic?
ANSWER:

QUESTION: What is the P-value?
ANSWER:

QUESTION: Based on the P-value and a confidence level of \(\alpha=.05\), what is your conclusion?
ANSWER:

QUESTION: What are the requirements for a hypothesis test for a proportion, and are they met?
ANSWER:

Exercise Habits

A health study surveyed 250 individuals about their exercise habits, and 140 reported exercising regularly. Construct a 97.25% confidence interval for the proportion of the general population that exercise regularly.

QUESTION: Interpret the confidence interval in context of the research question:
ANSWER:

QUESTION: Are the requirements for a valid confidence interval for \(p\) met?
ANSWER: