One and Two Sample Proportion Practice

One-Sample Test of Proportions

Teenage Smoking Habits

A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. A survey from 10 years ago claimed that 15% percent of all teenagers smoke.

A new 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 see if the new study shows a decrease in the percentage of teenagers who smoke from the 15% reported a decade ago.

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:

$$H_0:$$
$$H_a:$$ Choose your confidence level:

$$\alpha =0.$$

Perform the appropriate analysis:

prop.test(x = , n = , alternative = "")

Error in prop.test(x = , n = , alternative = ""): argument "x" is missing, with no default

Question: What is the P-value?
Answer:

Question: Based on your chosen $\alpha$ and P-value, what is your conclusion?
Answer:

Requirements

Recall that we must check that we have a big enough sample size to trust our p-value. To do this, we check that there are at least 10 expected “success” and “failures” for a given sample size, n:

Check:

$$np\ge 10$$

$$n(1-p)\ge 10$$

# Fill in n and the hypothesized p
n <- 
p <- 

n*p >= 10

Error: object 'n' not found

n*(1-p) >= 10

Error: object 'n' not found

QUESTION: Are the requirements for a hypothesis test satisfied?
ANSWER:

Confidence Interval

Create a 99% confidence interval for the true population proportion of teenagers who smoke?

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

Check Requirements

Recall that Confidence Intervals do not depend on a hypothesized proportion, so the requirements are a little different. For Confidence Intervals we check:

$$n\hat{p}\ge 10$$

$$n(1-\hat{p})\ge 10$$

# Fill in X and N and check that there are enough "successes" and "failures"

x <- 
n <- 
phat <- x/n

Error: object 'x' not found

n*phat >= 10

Error: object 'n' not found

n*(1-phat) >= 10

Error: object 'n' not found

2-Sample Proportion Tests

A Penny Saved?

A random sample of BYU-Idaho students was surveyed and asked if they were in favor of retaining the penny as a form of currency in the United States. Out of the 116 women surveyed, 80 said that they were in favor of retaining the penny as a form of currency. Of the 137 men surveyed, 91 said that they were in favor of retaining the penny.

For these questions, let group 1 represent women and group 2 represent men.

Hypothesis Test

Test to see if there is a difference between the proportion of women who want to keep the penny and the proportion of men who want to keep the penny. Use a level of significance of $\alpha =0.05$.

State your null and alternative hypotheses (replace the question marks with the appropriate symbols):

$$H_0:p_{female}???p_{male}$$
$$H_a:p_{female}???p_{male}$$

Perform the appropriate test:

prop.test(x = c(), n = c(), alternative = "")

Error in complete.cases(x, n): no input has determined the number of cases

Question: What is the P-value?
Answer:

Question: Based on $\alpha =0.05$, state your conclusion in context of the research question:
Answer:

Confidence Interval

Create a 95% confidence interval for the difference in the proportion of females to males who prefer to keep the penny:

Question: Interpret the confidence interval in context of the research question:
Answer:

Test Requirements

Recall that the requirements for Hypothesis Testing and Confidence Intervals for 2-sample proportions are the same. We must check that there are more than 10 “successes” and “failures” in both samples:

$$n_1{\hat{p}}_1\ge 10$$ $$n_1(1-{\hat{p}}_1)\ge 10$$ $$n_2{\hat{p}}_2\ge 10$$ $$n_2(1-{\hat{p}}_2)\ge 10$$

Use the following calculator to check the above requirements:

# All must be true:

x1 <- 
n1 <- 
phat1 <- x1/n1

Error: object 'x1' not found

n1*phat1 >= 10

Error: object 'n1' not found

n1*(1-phat1) >=10

Error: object 'n1' not found

x2 <- 
n2 <- 
phat2 <- x2 / n2

Error: object 'x2' not found

n2*phat2 >= 10

Error: object 'n2' not found

n2*(1-phat2) >=10

Error: object 'n2' not found

Question: Are the requirements satisfied for assuming normality?
Answer:

Couples Retreat?

A marriage counselor conducted a study of couples, categorizing each of the couples as “communicative” or “non-communicative”. Among other things, the counselor wanted to see whether the percentage of communicative couples whose marriage ended in separation or divorce was less than the percentage of non-communicative couples whose marriage ended in separation or divorce.

Of the 97 couples classified as “communicative”, 17 ended in separation or divorce. Of the 56 couples classified as “non-communicative”, 13 ended in separation or divorce.

Hypothesis Test

Construct a null and alternative hypothesis for the study:

$$H_0:$$
$$H_a:$$

Perform the appropriate analysis:

prop.test()

Error in prop.test(): argument "x" is missing, with no default

Question: What is the P-value?
Answer:

Question: Based on your decision rule, state your conclusion in context of the research question:
Answer:

Confidence Interval

Create a 95% confidence interval for the difference in the proportion of divorces between communicative and non-communicative couples:

Question: Interpret the confidence interval in context of the question:
Answer:

Test Requirements

Use the following calculator to check the requirements for the hypothesis test and confidence interval:

# All must be true:

x1 <- 
n1 <- 
phat1 <- x1/n1

Error: object 'x1' not found

n1*phat1 >= 10

Error: object 'n1' not found

n1*(1-phat1) >=10

Error: object 'n1' not found

x2 <- 
n2 <- 
phat2 <- x2 / n2

Error: object 'x2' not found

n2*phat2 >= 10

Error: object 'n2' not found

n2*(1-phat2) >=10

Error: object 'n2' not found

QUESTION: Are the requirements for the hypothesis test and confidence interval satisfied?
ANSWER: