91Ó°ÊÓ

Refer to Exercise \(8.3 .\) What effect does a larger population variance have on the margin of error?

Refer to Exercise 8.5 . What effect does an increased sample size have on the margin of error?

A random sample of \(n=50\) observations from a quantitative population produced \(\bar{x}=56.4\) and \(s^{2}=2.6 .\) Give the best point estimate for the population mean \(\mu,\) and calculate the margin of error.

An increase in the rate of consumer savings is frequently tied to a lack of confidence in the economy and is said to be an indicator of a recessional tendency in the economy. A random sampling of \(n=200\) savings accounts in a local community showed a mean increase in savings account values of \(7.2 \%\) over the past 12 months, with a standard deviation of \(5.6 \% .\) Estimate the mean percent increase in savings account values over the past 12 months for depositors in the community. Find the margin of error for your estimate.

A random sample of \(n=300\) observations from a binomial population produced \(x=263\) successes. Find a \(90 \%\) confidence interval for \(p\) and interpret the interval.

Acid rain, caused by the reaction of certain air pollutants with rainwater, appears to be a growing problem in the northeastern United States. (Acid rain affects the soil and causes corrosion on exposed metal surfaces.) Pure rain falling through clean air registers a pH value of 5.7 (pH is a measure of acidity: 0 is acid; 14 is alkaline). Suppose water samples from 40 rainfalls are analyzed for \(\mathrm{pH}\) and \(\bar{x}\) and \(s\) are equal to 3.7 and \(.5,\) respectively Find a \(99 \%\) confidence interval for the mean \(\mathrm{pH}\) in rainfall and interpret the interval. What assumption must be made for the confidence interval to be valid?

What is normal, when it comes to people's body temperatures? A random sample of 130 human body temperatures, provided by Allen Shoemaker \(^{9}\) in the Journal of Statistical Education, had a mean of 98.25 degrees and a standard deviation of 0.73 degrees. a. Construct a \(99 \%\) confidence interval for the average body temperature of healthy people. b. Does the confidence interval constructed in part a contain the value 98.6 degrees, the usual average temperature cited by physicians and others? If not, what conclusions can you draw?

Independent random samples were selected from populations 1 and 2 . The sample sizes, means, and variances are as follows: $$\begin{array}{lcc} & \multicolumn{2}{c} {\text { Population }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 35 & 49 \\\\\text { Sample Mean } & 12.7 & 7.4 \\\\\text { Sample Variance } & 1.38 & 4.14\end{array}$$ a. Find a \(95 \%\) confidence interval for estimating the difference in the population means \(\left(\mu_{1}-\mu_{2}\right) .\) b. Based on the confidence interval in part a, can you conclude that there is a difference in the means for the two populations? Explain.

A small amount of the trace element selenium, \(50-200\) micrograms \((\mu \mathrm{g})\) per day, is considered essential to good health. Suppose that random samples of \(n_{1}=n_{2}=30\) adults were selected from two regions of the United States and that a day's intake of selenium, from both liquids and solids, was recorded for each person. The mean and standard deviation of the selenium daily intakes for the 30 adults from region 1 were \(\bar{x}_{1}=167.1\) and \(s_{1}=24.3 \mu \mathrm{g}\) respectively. The corresponding statistics for the 30 adults from region 2 were \(\bar{x}_{2}=140.9\) and \(s_{2}=\) 17.6. Find a \(95 \%\) confidence interval for the difference in the mean selenium intakes for the two regions. Interpret this interval.

A study was conducted to compare the mean numbers of police emergency calls per 8 -hour shift in two districts of a large city. Samples of 100 8-hour shifts were randomly selected from the police records for each of the two regions, and the number of emergency calls was recorded for each shift. The sample statistics are listed here: $$\begin{array}{lcc} & \multicolumn{2}{c} {\text { Region }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 100 & 100 \\\\\text { Sample Mean } & 2.4 & 3.1 \\\\\text { Sample Variance } & 1.44 & 2.64\end{array}$$ Find a \(90 \%\) confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts of the city. Interpret the interval.