91Ó°ÊÓ

State the circumstances under which we construct a \(t\) interval. What are the circumstances under which we construct a Z-interval?

Why does the margin of error increase as the level of confidence increases?

Discuss the similarities and differences between the standard normal distribution and the \(t\) -distribution.

Suppose a professor in a class of 20 students wants to estimate the mean pulse rate of students prior to the final exam. Would it make sense for the professor to construct a \(90 \%\) confidence interval about the population mean? Explain.

(a) Find the \(t\) -value such that the area in the right tail is 0.10 with 25 degrees of freedom. (b) Find the \(t\) -value such that the area in the right tail is 0.05 with 30 degrees of freedom. (c) Find the \(t\) -value such that the area left of the \(t\) -value is 0.01 with 18 degrees of freedom. [Hint: Use symmetry. (d) Find the critical \(t\) -value that corresponds to \(90 \%\) confidence. Assume 20 degrees of freedom.

Construct a confidence interval of the population proportion at the given level of confidence. $$x=400, n=1200,95 \% \text { confidence }$$

A simple random sample of size \(n\) is drawn from a population that is normally distributed. The sample mean, \(\bar{x}\), is found to be \(50,\) and the sample standard deviation, \(s,\) is found to be \(8 .\) (a) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) (b) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(15 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

Construct a confidence interval of the population proportion at the given level of confidence. $$x=540, n=900,96 \% \text { confidence }$$

A simple random sample of size \(n\) is drawn. The sample mean, \(\bar{x},\) is found to be \(18.4,\) and the sample standard deviation, \(s\), is found to be \(4.5 .\) (a) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 35 (b) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(50 .\) How does increasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(99 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(35 .\) Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error, \(E ?\) (d) If the sample size is \(n=15,\) what conditions must be satisfied to compute the confidence interval?

Construct the appropriate confidence interval. In a random sample of 40 felons convicted of aggravated assault, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct and interpret a \(95 \%\) confidence interval for the mean length of sentence for an aggravated assault conviction. (Source: Based on data obtained from the U.S. Department of Justice.)