Unit 3 check: Inference for Categorical Data: Proportions

An orchard's records show that 15% of its apples are bruised. Inspectors take random samples of 500 apples, and the standard deviation of the sampling distribution of the sample proportion of bruised apples is about 0.016. Which is the best interpretation of 0.016?

A researcher randomly samples 750 adults in Ohio and asks whether they have a library card. Which is the best description of the parameter being estimated?

A consumer group writes the hypotheses $H_0:p\le 0.05$ and $H_a:p>0.05$, where $p$ is the proportion of a brand's smoke detectors that fail a safety check. Which value of $p$ is used to carry out the test?

A city says that 70% of its residents recycle. A researcher suspects the proportion is lower. In a random sample of 300 residents, 195 recycle, so $\hat{p}=0.65$. For the test of $H_0:p=0.70$ against $H_a:p<0.70$, the p-value is 0.029. Which is a correct interpretation of the p-value?

A smartwatch company claims that more than 30% of its buyers use the sleep tracker every night. In a random sample of 500 buyers, 165 do. The test of $H_0:p=0.30$ against $H_a:p>0.30$ gives $z=1.46$ and a p-value of 0.072. At $\alpha =0.05$, which is the correct decision and conclusion?

A test of $H_0:p=0.50$ against $H_a:p<0.50$ uses $\alpha =0.05$. The power of the test against a true proportion of 0.40 is 0.72. If the true proportion really is 0.40, what is the probability that the test makes a Type II error?

At one large company, 35% of employees work fully remote. At a second large company, 25% do. Random samples of 100 employees are taken from each company. What is the probability that ${\hat{p}}_1-{\hat{p}}_2$, the first company's sample proportion minus the second's, is 0.20 or more?

Researchers take a random sample of teens and a separate random sample of adults in a large city. A 90% confidence interval for $p_T-p_A$, the proportion of teens minus the proportion of adults who used a ride-share app in the past month, is $(0.08,0.17)$. Which interpretation is correct?

In an experiment, 24 of 120 seedlings given a new fertilizer and 15 of 100 seedlings given the standard fertilizer wilt within a week. A two-sample z-test will test whether the proportions that wilt differ. Which is the correct check of the normality condition for this test?

A two-sample z-test of whether the proportion of adults who prefer paper books differs between two large cities, using a random sample from each city, gives a p-value of 0.12. At $\alpha =0.05$, which conclusion is correct?

A chi-square test for independence uses a two-way table with 3 rows and 4 columns. The smallest observed count in the table is 3, and the smallest expected count is 6.2. Is the expected counts condition met?

A random sample of 400 students at a university is classified by handedness and by preferred sport. A chi-square test for independence gives a p-value of 0.31. At $\alpha =0.05$, which conclusion is correct?