If a random sample of size $n$ is drawn from a population that is exactly normal with mean $\mu$ and variance ${\sigma }^2$, which statement is correct about the sampling distribution of the sample mean $\bar{X}$?

The distribution of the time to complete a vehicle safety inspection is skewed to the right. Records show the mean inspection time is $9.8$ minutes with a standard deviation of $4.1$ minutes. What sample size is commonly required so that the sampling distribution of the sample mean can be approximated by a normal distribution for probability calculations?

A researcher wants to test the effect of a memory training program. Before the program, a participant could recall $12$ items from a list. After the program, the participant was tested on $10$ different lists, with the following numbers of items recalled: $15,17,16,18,14,19,16,17,15,18$. To estimate the mean post-program recall score, the researcher plans to generate $5000$ independent bootstrap samples of size $n=10$ with replacement. What is the procedure for building the bootstrap distribution of the sample mean?

The mean daily rainfall in a rainforest region is normally distributed with a mean of $18.2\text{ mm}$ and a standard deviation of $6.3\text{ mm}$. If a sample of $10$ days is randomly chosen, what is the probability that the sample mean rainfall is between $14.4$ and $21.7\text{ mm}$?

A sociologist is constructing a confidence interval for the average number of hours worked per week using a sample of $70$ individuals from a population with a left-skewed distribution. The population standard deviation is not provided. Which distribution should be used?

If a researcher constructs a 90% confidence interval for the mean income of a city, what does the 90% confidence level mean?

A certain stock's daily percent return on Fridays has a mean of $3.12\%$ and a standard deviation of $41.25\%$. If random samples of $40$ days are selected and the mean return for each sample is calculated, what is the probability that a sample mean is between $-5\%$ and $10\%$?

A tech company is developing a new fast-charging battery and wants to estimate the average time it takes to charge fully. The company requires the estimate to be accurate within $0.40$ hours of the true population mean with $95\%$ confidence. If the population standard deviation is known to be $1.6$ hours, what is the minimum number of batteries they must test to achieve this precision?

The annual salaries (in units of dollars) for $10$ randomly chosen dental hygienists are listed below. Assume the population is normally distributed.

Construct a $95\%$ confidence interval for the population mean annual salary for dental hygienists. Round your answer to the nearest whole number.

In the context of determining minimum sample size for estimating a mean, what is the effect on the sample size if the margin of error is reduced, while all other parameters remain unchanged?

A researcher collects a simple random sample of waiting times in minutes for a popular amusement park ride. A confidence level of $95\%$ was used. The researcher used a calculator for analysis, as shown in the figure.

At the confidence level of $95\%$, what is the critical value $t_{\frac{\alpha }{2}}$?

Suppose you’re constructing a confidence interval to estimate the average time spent on a mobile app. If the sample size, sample mean, and confidence level remain the same, what effect does an increase in the population standard deviation (e.g., from $5$ minutes to $10$ minutes) have on the width of the confidence interval? Briefly explain your answer.

A researcher collects a simple random sample of waiting times in minutes for a popular amusement park ride. A confidence level of $95\%$ was used. The researcher used a calculator for analysis, as shown in the figure.

Provide a statement that correctly interprets the confidence interval displayed.

A survey records the time, in minutes past $7:00$ A.M., that a random sample of $30$ students wake up for an early class. The population standard deviation is known to be $25$ minutes. Calculate the point estimate of the population mean, $\mu$, and the margin of error for a $90\%$ confidence interval.

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A wellness researcher is studying the average daily screen time of adults. In a previous analysis, a random sample of $10$ adults showed a mean daily screen time of $5.38$ hours with a standard deviation of $0.43$ hours. The researcher claims that the population mean daily screen time is $5.2$ hours. Assuming the population of screen times are normally distributed, determine whether the calculated $t$-value falls between $-{𝑡}_{0.05}$ and $t_{0.05}$ at the $90\%$ confidence level.