Unit 7 test: Inference for Means

A one-sample $t$ interval is built from a random sample of $22$ observations. How many degrees of freedom does it use?

Why do inference procedures for a population mean usually use $t$ critical values instead of $z$ critical values?

Use the table excerpt to find the critical value $t^*$ for a 98% confidence interval based on a random sample of $13$ observations.

A random sample of $20$ commuters reported a mean one-way commute of $\bar{x} = 15.8$ miles with $s = 3.4$ miles. A dotplot shows no strong skew or outliers. Construct a 95% confidence interval for the true mean commute distance ($t^* = 2.093$). Enter the endpoints to 2 decimal places, separated by a comma.

A researcher reports a 90% confidence interval for the mean weight of a species of frog. What does "90% confidence" mean?

A 95% confidence interval for the mean price of a used textbook at a college bookstore is $(20.6,\ 26.2)$ dollars. What is the margin of error?

A cereal maker says its boxes contain a mean of $500$ grams. A consumer group suspects the mean is actually higher than labeled. A random sample of $40$ boxes has $\bar{x} = 508.6$ g and $s = 24$ g. Find the test statistic for $H_0: \mu = 500$, to 2 decimal places.

For the cereal test in the previous problem, $H_a: \mu > 500$ and $df = 39$. Find the P-value to 4 decimal places.

Using the cereal test (P-value $\approx 0.0145$) at $\alpha = 0.05$, which conclusion is correct?

A two-sided one-sample $t$ test of $H_0: \mu = 72$ gives a P-value of $0.18$. Which 95% confidence interval could come from the same data?

Ten randomly selected runners ran a mile on a track wearing old shoes and wearing new shoes, in random order. The mean of the differences (old − new, in seconds) was $\bar{x}_{\text{diff}} = 1.8$ with $s_{\text{diff}} = 2.2$. Find the paired $t$ statistic for $H_0: \mu_{\text{diff}} = 0$, to 2 decimal places.

For the runner study in the previous problem, which statement about the conditions is correct?

A random sample of $25$ customers at Store A spent a mean of $42.30 with $s = 5.1$ dollars. An independent random sample of $28$ customers at Store B spent a mean of $39.00 with $s = 6.3$ dollars. Find the two-sample $t$ statistic for $H_0: \mu_A - \mu_B = 0$, to 2 decimal places.

For the store comparison in the previous problem, the alternative is $H_a: \mu_A - \mu_B \ne 0$ and technology reports $df \approx 50.5$. Find the P-value to 4 decimal places.

A school randomly assigns $60$ volunteers to either a new vocabulary app or traditional flashcards, $30$ per group. A two-sample $t$ test finds that the app group's mean score is significantly higher ($\text{P-value} = 0.008$). Which conclusion is best?