AP Statistics – Inference Practice

In a random sample of $400$ adults from a city, $120$ said they cycle to work at least once a week.(a) Verify that the conditions for a one-sample $z$-interval for a proportion are met.(b) Construct and interpret a $95$ percent confidence interval for the proportion of all adults in the city who cycle to work at least once a week.

A random sample of $25$ packets of coffee has mean mass $52.4$ g and standard deviation $6.8$ g. A dotplot of the sample shows no strong skew or outliers.(a) Explain why a $t$-procedure is appropriate.(b) Construct a $90$ percent confidence interval for the mean mass of all packets.

A polling organisation wants to estimate the proportion of voters who support a proposal to within $\pm 0.03$ with $95$ percent confidence. No prior estimate is available.Find the minimum sample size required.

A company claims that $60$ percent of its customers renew their subscription. In a random sample of $400$ customers, $210$ renewed.(a) State the hypotheses for a test of the company’s claim.(b) Calculate the test statistic and $P$-value.(c) State your conclusion at $\alpha = 0.05$.

A machine is set to fill bottles with $500$ mL. A random sample of $36$ bottles has mean $492$ mL and standard deviation $18$ mL.(a) Calculate the $t$ statistic for testing $H_0: \mu = 500$ against $H_a: \mu < 500$.(b) Find the $P$-value and state a conclusion at the $1$ percent significance level.

In a randomised experiment, $150$ patients received a new drug and $150$ received a placebo. $45$ of the drug group and $30$ of the placebo group reported improvement.Carry out a test at $\alpha = 0.05$ to determine whether the drug increases the proportion of patients who improve.

Twenty students took a reaction-time test before and after drinking a caffeinated drink. For each student the difference (before minus after, in hundredths of a second) was computed. The mean difference was $2.3$ with standard deviation $4.1$; a plot of the differences was roughly symmetric.(a) Explain why a paired $t$-test is appropriate rather than a two-sample $t$-test.(b) Test whether caffeine reduces reaction time, using $\alpha = 0.05$.

A six-sided die is rolled $120$ times. The observed frequencies of $1, 2, 3, 4, 5, 6$ are $15, 22, 18, 25, 20, 20$.(a) State the hypotheses for a chi-square goodness-of-fit test.(b) Calculate the chi-square statistic and its degrees of freedom.(c) Find the $P$-value and state a conclusion at $\alpha = 0.10$.

A random sample of $120$ shoppers was classified by gender and preferred store layout (A, B or C). Males: A $30$, B $20$, C $10$. Females: A $20$, B $30$, C $10$.(a) Find the expected count for males preferring layout A.(b) Calculate the chi-square statistic and degrees of freedom for a test of independence.(c) State the $P$-value and conclusion at $\alpha = 0.05$.

Computer output for a regression of exam score on hours studied ($n = 20$ students) gives slope $b = 2.45$ with standard error $0.62$. Conditions for inference are satisfied.(a) Construct a $95$ percent confidence interval for the population slope $\beta$.(b) Test $H_0: \beta = 0$ against $H_a: \beta eq 0$ and interpret the result.

Two teaching methods were compared. Method A: $n = 30$, $\bar{x} = 78.2$, $s = 8.1$. Method B: $n = 28$, $\bar{x} = 74.5$, $s = 9.4$. Students were randomly assigned to methods.(a) Calculate the standard error of $\bar{x}_A – \bar{x}_B$.(b) Carry out a two-sample $t$-test (using the conservative df $= 27$ or technology) to determine whether the methods differ in mean score, at $\alpha = 0.05$.(c) Describe a Type I error in this context.

Historically $20$ percent of applicants to a programme are accepted. A new admissions officer accepts $16$ of a random sample of $50$ applicants from this year.(a) Check whether a one-proportion $z$-test is appropriate.(b) Test whether the acceptance rate has increased, at $\alpha = 0.05$.(c) Explain what is meant by the power of this test, and give one way to increase it.