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1 MATH 1342.P04 CHAPTERS 7, 8 & 9 FALL 2009 NAME SHOW ALL WORK FOR CREDIT!!! 1. The U.S. Bureau of Labor Statistics conducts periodic surveys to collect information on the labor market. According to the bureau, workers in the private sector earned an average of $18.15 per hour in May, Assume that the current hourly wages for all workers in the private sector have a normal distribution and that the standard deviation is $2.15. a. If a worker in the private sector is selected at random, what is the probability that he/she earns less than $16.50 per hour? b. If a worker in the private sector is selected at random, what is the probability that he/she earns between $12 and $15 per hour? c. If a worker in the private sector is selected at random, what is the probability that he/she earns less than $20 per hour? d. What is the hourly wage corresponding to the 10th percentile?
2 MATH 1342.P04 PAGE 2 2. According to the records of an electric company serving the Boston area, the mean electric consumption for all households during the winter is 1650 kilowatts per hour. Assume that the monthly electric consumption during winter by all households in this area has a normal distribution and that the standard deviation is 320 kilowatt hours. a. If a household is selected at random, find the probability that the household uses more than 2000 kilowatts per hour during the winter. b. The electric company sent a notice to Richard Blaine informing him that 90% of the households use less electricity per hour in the winter than he does. What is Richard's hourly electric consumption? 3. Total serum cholesterol levels for individuals 65 years of age and older are assumed to follow a normal distribution with a mean of 182 mg/dl and a standard deviation of 14.7 mg/dl. a. If an individual that is 65 years of age and older is selected at random, what is the probability he/she will have a cholesterol level between 170 mg/dl and 187 mg/dl? b. If a random sample of 25 individuals 65 years or older is selected, what is the probability that the mean total serum cholesterol level is less than 178 mg/dl?
3 MATH 1342.P04 PAGE 3 4. The GPAs of all students enrolled at a large university have an approximate normal distribution with a mean of 3.02 and a standard deviation of Find the probability that the mean GPA of a random sample of 30 students selected form the university is 3.10 or higher. 5. According to Labor Canada, 29.2% of Canada's civilian workers were members of labor unions in Suppose this result holds true for the current population on Canadian civilian workers. If a random sample of 600 Canadian civilian workers, what is the probability that the sample proportion of workers are union members is greater than 33%. 6. A study is conducted to determine if people with glaucoma have higher blood pressure than average. A sample of 21 people with glaucoma resulted in a mean systolic blood pressure of 140 mm Hg with a standard deviation of 25 mm Hg. Assume that the systolic blood pressure for people with glaucoma is normally distributed. Construct a 95% confidence interval for the mean systolic blood pressure among people with glaucoma.
4 MATH 1342.P04 PAGE 4 7. ACME Lumber Company wants to determine the load required to pull apart pieces of Douglas fir that are 4 inches long and 1.5 inches square. Researchers performed a stress test to determine the strength (measured in pounds) with a sample of 20 pieces of wood that resulted in the following data: Assuming that the strength of pieces of wood like these follows a normal distribution. a. Construct a 90% confidence interval for the true mean load required to pull the wood apart. b. In a sentence, explain your resulting confidence interval from part a. 8. In a poll of 2003 U.S. adults conducted by Hart & Teeter for The Wall Street Journal/NBC News, 57% of adults said that public education needs to be improved. Construct a 95% confidence interval for the proportion of all U.S. adults who hold this opinion.
5 MATH 1342.P04 PAGE 5 9. A pollster is hired to determine the percentage of voters favoring the Republican gubernatorial nominee. How large a sample is needed if the pollster requires 95% confidence and the estimated value should be within two percentage points of the true proportion? 10. Because cardiac deaths appear to increase after heavy snowfalls, an experiment was designed to compare cardiac demands of snow shoveling to those of using an electric snow thrower. Ten subjects cleared tracts by shoveling the snow and their maximum heart rates (beats per minute) were recorded during the activity, and the standard deviation heart rate was found to be 15 bpm. Construct a 95% confidence interval estimate for the population standard deviation.
6 MATH 1342.P04 PAGE 6 Formulas The Mean and Standard Deviation of the Sampling Distribution of B Suppose that a simple random sample of size 8 is drawn from a large population with mean. and standard deviation 5. The sampling distribution of B will have mean. B œ. and standard 5 deviation 5B œ. The standard deviation of the sampling distribution of B, 5B, is called the È8 standard error of the mean. The Shape of the Sampling Distribution of B if \ is Normal If a random variable \ is normally distributed, the distribution of the sample mean, B, is normally distributed. The Central Limit Theorem Regardless of the shape of the underlying population, the sampling distribution of approximately normal as the sample size, 8, increases. B becomes Sampling Distribution of :s For a simple random sample of size 8 with population proportion :, The shape of the sampling distribution of ^: is approximately normal provided 8: " : "!. The mean of the sampling distribution of ^: is. :s œ :. The standard deviation of the sampling distribution of ^: is œ :" : 5 :s Ê. 8 Sample Size Needed for Estimating the Population Proportion : The sample size required to obtain a " α "!!% confidence interval for : with margin of error I is given by 8œ: ; s s D αî# # Š I (rounded up to the next integer), where ^: is a prior estimate of :. If a prior estimate of : is unavailable, the sample size required is rounded up to the next integer. αî# 8 œ!þþ#& Š D I A " α "!!% Confidence Interval about 5 # If a simple random sample of size 8 is taken from a normal populations with mean. and # standard deviation 5, then a " α "!!% confidence interval about 5 is given by 8 " = # Lower bound: Upper bound: # ;# V 8 " = # ;# P
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