91Ó°ÊÓ

Consider a binomial experiment with \(n\) trials and \(r\) successes. For a test for a proportion \(p,\) what is the formula for the \(z\) value of the sample test statistic? Describe each symbol used in the formula.

If sample data is such that for a one-tailed test of \(\mu\) you can reject \(H_{0}\) at the \(1 \%\) level of significance, can you always reject \(H_{0}\) for a two-tailed test at the same level of significance? Explain.

For a Student's \(t\) distribution with \(d . f .=16\) and \(t=-1.830\), (a) find an interval containing the corresponding \(P\) -value for a two-tailed test. (b) find an interval containing the corresponding \(P\) -value for a left-tailed test.

Sociology: Crime Rate Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980 s and 1990 s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, \(70 \%\) of all arrests are of males aged 15 to 34 years (Source: True Odds by J. Walsh, Merritt Publishing). Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 32 arrests last month, 24 were of males aged 15 to 34 years. Use a \(1 \%\) level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from \(70 \%.\)

College Athletics: Graduation Rate Women athletes at the University of Colorado, Boulder, have a long-term graduation rate of \(67 \%\) (Source: Chronicle of Higher Education). Over the past several years, a random sample of 38 women athletes at the school showed that 21 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the University of Colorado, Boulder, is now less than \(67 \% ?\) Use a \(5 \%\) level of significance.

Suppose you want to test the claim that a population mean equals \(40 .\) (a) State the null hypothesis. (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from \(40 .\) (c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may exceed \(40 .\) (d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be less than \(40 .\)

Unfortunately, arsenic occurs naturally in some ground water (Reference: Union Carbide Technical Report \(K / U R-1\) ). A mean arsenic level of \(\mu=8.0\) parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is tested on a regular basis for arsenic. A random sample of 37 tests gave a sample mean of \(\bar{x}=7.2 \mathrm{ppb}\) arsenic, with \(s=1.9 \mathrm{ppb} .\) Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use \(\alpha=0.01.\)

USA Today reported that the state with the longest mean life span is Hawaii, where the population mean life span is 77 years. A random sample of 20 obituary notices in the Honolulu Advertizer gave the following information about life span (in years) of Honolulu residents: \(\begin{array}{llllllll}72 & 68 & 81 & 93 & 56 & 19 & 78 & 94 & 83 & 84\end{array}\) \(\begin{array}{llllllll}77 & 69 & 85 & 97 & 75 & 71 & 86 & 47 & 66 & 27\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}=71.4\) years and \(s \approx 20.65\) years. ii. Assuming that life span in Honolulu is approximately normally distributed, does this information indicate that the population mean life span for Honolulu residents is less than 77 years? Use a \(5 \%\) level of significance.