7. Sampling Distributions & Confidence Intervals: Mean

For a two-tailed hypothesis test using a 5% significance level, what is the critical value of $z$?

According to the law of large numbers, as the sample size increases, which of the following statements is true about the $\mathrm{sample mean}$?

Which of the following formulas correctly calculates the standard error of the mean for a data set with sample standard deviation $s$ and sample size $n$?

Which of the following would not cast doubt on the usefulness of sample data in constructing a confidence interval?

Suppose the waiting times for patients needing emergency service are normally distributed with a mean of $\mu =12$ minutes and a standard deviation of $\sigma =3$ minutes. What proportion of patients wait $9$ minutes or less?

In the context of hypothesis testing, which of the following gives the probability of making a Type I error?

"A second method for finding a bootstrap confidence interval is called the Bootstrap t-Method. This method requires estimating the standard error of the estimate (such as the standard error of the mean) from the bootstrap sample. For any given set of estimates of a parameter, the standard error of the estimate is found by determining the sample standard deviation of the B bootstrap estimates. For example, in Example 1, we found 2000 estimates of the sample mean mpg. The standard deviation of these 2000 sample means is found to be 0.580. The standard deviation of the 16 observations is 2.38, so

s / sqrt(n) = 2.38 / sqrt(16) = 0.595,

is close to the standard error of the estimate found from the bootstrap samples. The estimate of the standard error from Figure 28 in Example 2 is 0.560. We can use an estimate of the standard error (SE_est) along with critical values from Student's t-distribution (Table VII) to construct confidence intervals as follows:

statistic ± t_(alpha/2) * SE_est

For example, in Example 1 we know x̄ = 28.1 and n = 16, so for a 95% confidence interval, t_0.025 = 2.131 using 15 degrees of freedom. Using the standard error estimate from Example 1, we find the lower bound of the confidence interval to be 28.1 – 2.131(0.580) = 26.86 mpg and the upper bound to be 28.1 + 2.131(0.580) = 29.34.

pH of Rain Revisited See Problem 7. Use the Bootstrap t-Method to find a 95% confidence for the mean pH of rainwater in Tucker County, West Virginia."

Which of the following best describes what the Central Limit Theorem states in the context of confidence intervals?

Make a 90% confidence interval for a parameter, y, with point estimate $ŷ=-1.5$, & margin of error $E=3.25$.

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.98

Find the critical value, $z_{\frac{\alpha }{2}}$, for a 80% confidence interval.

Suppose a 95% confidence interval for the mean difference in blood pressure between a treatment group and a control group is $(-2,5)$. Based on this interval, what does the confidence interval suggest about the effectiveness of the treatment?

Given a time-series trend equation of $25.3+2.1x$, what is the forecasted value for period $7$?

Which of the following is not a characteristic of the distribution of sample means?

Based on the bar chart showing the $95\%$ confidence intervals for the mean test scores of four different classes, which of the following is an accurate conclusion?