A manufacturer claims that the average battery life of their AA batteries is at least $5.2$ hours. A consumer group tests a random sample of $12$ batteries and finds a mean battery life of $4.9$ hours. Assume the population standard deviation is $0.6$ hours, and battery life is normally distributed. At the $0.01$ significance level, can you reject the manufacturer’s claim?

A researcher claims that the mean weight of apples in a storage bin is $180$ grams. To test this claim, a quality control officer collects a random sample of $12$ apples, and records their weights (in grams) as:

$178,182,175,180,176,179,181,183,177,179,180,178$

Assuming that the weights are normally distributed, and the sample standard deviation is approximately $2.38$ grams, determine whether there is enough evidence at the $0.05$ significance level to reject the researcher’s claim that the mean weight is $180$ grams.

A tech company claims that the mean battery life of its new smartwatch is at least $24$ hours. A skeptical reviewer believes the battery life is less than $24$ hours. A random sample of 10 smartwatches resulted in a mean battery life of 24 hours. After simulating $2000$ randomizations under the null hypothesis, $70$ simulated means are as extreme or more extreme than $24$ hours.

Use a randomization procedure to test the reviewer's claim at the $0.05$ significance level.

In a study of $2000$ patients, researchers recorded both the actual and estimated number of hours slept per night. For each patient, the difference (actual minus estimated) was computed. The sample mean of the differences is $-0.3$ hours, and the sample standard deviation is $1.2$ hours. Test whether there is a significant difference between actual and estimated sleep hours at the $0.05$ significance level. What is the $p$-value for this hypothesis test?

In $2012$, the mean zinc concentration in a lake was reported as $0.035$ $\text{mg/L}$. In $2021$, a random sample of $9$ measurements yielded: $0.032$, $0.034$, $0.036$, $0.033$, $0.035$, $0.037$, $0.034$, $0.036$, $0.033$ $\text{mg/L}$. Assume normality and no outliers. At the $\alpha =0.05$ significance level, is there evidence that the mean zinc concentration has changed since $2012$?

A researcher claims that the mean annual salary for a certain profession is at least $\mathrm{\backslash (}60,000$. At the $0.05$ significance level, test this claim using the following sample statistics:

$\bar{x}=\mathrm{\backslash )}58,800$

$s=\$2,400$

$n=16$

Assume the population is normally distributed. What is the result of the hypothesis test?

A researcher wants to determine if there is a significant difference in math test performance between students who use digital textbooks and those who use printed textbooks. The following data were collected:

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$.

At the $0.05$ significance level, test the claim that there is no significant difference in mean test performance between the two groups.

A university administrator claims that the average time to graduate is not greater than $4.5$ years. A sample of $40$ recent graduates shows a mean graduation time of $4.7$ years with a standard deviation of $0.8$ years. At $\alpha =0.05$, is there sufficient evidence to reject the administrator's claim? Assume the population is normally distributed.

A psychologist is studying the effect of a stress management workshop on anxiety levels. Eight participants took an anxiety test before and after attending the workshop. Their scores are as follows:

a. Using a $0.01$ significance level, test the claim that the mean anxiety score before the workshop is higher than after the workshop.

b. Construct the $99\%$ confidence interval for the mean difference. What does this interval indicate about anxiety change?

c. What can you conclude about the effectiveness of the workshop?

A university historically has a course completion rate of $61\%$. They pilot a peer-mentoring intervention with $250$ students and observe that $140$ students complete the course. Identify the variable of interest and classify its type.