BackDetermine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.
The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)
c. Would it be unusual for the population mean to be under 575? Explain.
The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)
d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.
Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
b. In a random sample of 15 cereal boxes, the mean weight was 11.89 ounces. Assume the weights of the cereal boxes are normally distributed and the population standard deviation is 0.05 ounce. Construct a 90% confidence interval for the population mean.
Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (c) construct a 98% confidence interval for the population mean.
[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)
Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.
a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.
Ages of College Students An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages is normally distributed.
a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 1.6 years.
In a random sample of 36 top-rated roller coasters, the average height is 165 feet and the standard deviation is 67 feet. Construct a 90% confidence interval for μ. Interpret the results. (Source: POP World Media, LLC)
The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.
The confidence interval for Year 2 is much larger than that for the other years. What do you think may have caused this larger confidence level?
The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.
What can the water department do to decrease the size of the confidence intervals, regardless of the amount of variance in cyanide levels?
In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?
c=0.95, x̅=3.46, s=1.63, n=16
In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?
c=0.90, x̅=8.21, σ=0.62, n=8
[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)
48,69446,85642,91261,67271,11254,861
69,45471,84159,75169,61254,28452,166
66,36048,16465,27235,25061,12765,397
58,92558,91659,01753,07045,19969,941
69,49257,08553,82952,69268,29853,792
Construct a 95% confidence interval for the population mean annual earnings for locksmiths.
Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.
Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.
Repeat Exercise 26 for samples of size 72 and 108. What happens to the mean and the standard deviation of the distribution of sample means as the sample size increases?