Statistical Inference About Means and Proportions with Two Populations

Injuries to Major League Baseball players have been increasing in recent years. For the period 1992 to 2001 , league expansion caused Major League Baseball rosters to increase 15%15% However, the number of players being put on the disabled list due to injury increased 32%32% over the same period (USA Today, July 8,2002 ). A research question addressed whether Major League Baseball players being put on the disabled list are on the list longer in 2001 than players put on the disabled list a decade earlier. a. Using the population mean number of days a player is on the disabled list, formulate null and alternative hypotheses that can be used to test the research question. b. Assume that the following data apply: a. Using the population mean number of days a player is on the disabled list, formulate null and alternative hypotheses that can be used to test the research question. b. Assume that the following data apply: Sample sizeSample meanSample standard deviation2001Seasonn1=45x1=60dayss1=18days1992Seasonn2=38x2=51dayss2=15days2001Season1992SeasonSample sizen1=45n2=38Sample meanx1=60daysx2=51daysSample standard deviations1=18dayss2=15days What is the point estimate of the difference between population mean number of days on the disabled list for 2001 compared to 1992?1992? What is the percentage increase in the number of days on the disabled list? c. Use =.01.=.01. What is your conclusion about the number of days on the disabled list? What is the pp -value? d. Do these data suggest that Major League Baseball should be concerned about the situation?

Coastal areas of the United States including Cape Cod, the Outer Banks, the Carolinas, and the Gulf Coast had relatively high population growth rates during the 1990 s. Data were collected on residents living in the coastal communities as well as on residents living in noncoastal areas throughout the United States (USA Today, July 21, 2000). Assume that the following sample results were obtained on the ages of individuals in the two populations: Coastal Areas Noncoastal Areas n1=150x1=39.3yearss1=16.8yearsn2=175x2=35.4yearss2=15.2yearsn1=150n2=175x1=39.3yearsx2=35.4yearss1=16.8yearss2=15.2years Test the hypothesis of no difference between the two population means. Use =.05=.05 a. Formulate the null and alternative hypotheses. b. What is the value of the test statistic? c. What is the pp -value? d. What is your conclusion?

FedEx and United Parcel Service (UPS) are the world's two leading cargo carriers by volume and revenue (The Wall Street Journal, January 27, 2004). According to the Airports Council International, the Memphis International Airport (FedEx) and the Louisville International Airport (UPS) are two of the ten largest cargo airports in the world. The following random samples show the tons of cargo per day handled by these airports. Data are in thousands of tons. Memphis9.18.3Louisville4.72.215.19.15.04.18.86.04.22.610.05.83.33.47.512.15.57.010.59.3Memphis9.115.18.810.07.510.58.39.16.05.812.19.3Louisville4.75.04.23.35.52.24.12.63.47.0 a. Compute the sample mean and sample standard deviation for each airport. b. What is the point estimate of the difference between the two population means? Interpret this value in terms of the higher-volume airport and a comparison of the volume difference between the two airports. c. Develop a 95%95% confidence interval of the difference between the daily population means for the two airports.

The U.S. Department of Transportation provides the number of miles that residents of the 75 largest metropolitan areas travel per day in a car. Suppose that for a simple random sample of 50 Buffalo residents the mean is 22.5 miles a day and the standard deviation is 8.4 miles a day, and for an independent simple random sample of 40 Boston residents the mean is 18.6 miles a day and the standard deviation is 7.4 miles a day. a. What is the point estimate of the difference between the mean number of miles that Buffalo residents travel per day and the mean number of miles that Boston residents travel per day? b. What is the 95%95% confidence interval for the difference between the two population means?

Consider the following hypothesis test. H0:12=0Ha:120H0:12=0Ha:120 The following results are from independent samples taken from two populations. Sample 1n1=35x1=13.6s1=5.2Sample 2n2=40x2=10.1s2=8.5Sample 1Sample 2n1=35n2=40x1=13.6x2=10.1s1=5.2s2=8.5 a. What is the value of the test statistic? b. What is the degrees of freedom for the tt distribution? c. What is the pp -value? d. At =.05,=.05, what is your conclusion?

Consider the following data for two independent random samples taken from two normal populations. Sample 1Sample 210877138749689Sample 110713798Sample 2878469 a. Compute the two sample means. b. Compute the two sample standard deviations. c. What is the point estimate of the difference between the two population means? d. What is the 90%90% confidence interval estimate of the difference between the two population means?

During the 2003 season, Major League Baseball took steps to speed up the play of baseball games in order to maintain fan interest (CNN Headline News, September 30, 2003). The following results come from a sample of 60 games played during the summer of 2002 and a sample of 50 games played during the summer of 2003.2003. The sample mean shows the mean duration of the games included in each sample. 2002 Season n1=60n1=60 x1=2x1=2 hours ,52,52 minutes

2003 Season n2=50n2=50 x2=2x2=2 hours, 46 minutes a. A research hypothesis was that the steps taken during the 2003 season would reduce the population mean duration of baseball games. Formulate the null and alternative hypotheses. b. What is the point estimate of the reduction in the mean duration of games during the 2003 season? c. Historical data indicate a population standard deviation of 12 minutes is a reasonable assumption for both years. Conduct the hypothesis test and report the pp -value. At a .05 level of significance, what is your conclusion? d. Provide a 95%95% confidence interval estimate of the reduction in the mean duration of games during the 2003 season. e. What was the percentage reduction in the mean time of baseball games during the 2003 season? Should management be pleased with the results of the statistical analysis? Discuss. Should the length of baseball games continue to be an issue in future years? Explain.

The average expenditure on Valentine's Day was expected to be $100.89$100.89 (USA Today February 13,2006 ). Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 40 male consumers was $135.67$135.67, and the average expenditure in a sample survey of 30 female consumers was $68.64.$68.64. Based on past surveys, the standard deviation for male consumers is assumed to be $35,$35, and the standard deviation for female consumers is assumed to be $20$20 a. What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females? b. At 99%99% confidence, what is the margin of error? c. Develop a 99%99% confidence interval for the difference between the two population means.

The nation's 40,000 mortgage brokerages are some of the most profitable small businesses in the United States. These low-profile companies find loans for customers in exchange for commissions. Mortgage Bankers Association of America provides data on the average size of loans handled by mortgage brokerages (The Wall Street Journal, February 24, 2003). The CD file named Mortgage contains data from a sample of 250 loans made in 2001 and a sample of 270 loans made in 2002 that are consistent with these data. Based on historical loan data, the population standard deviations for the loan amounts can be assumed known at $55,000$55,000 in 2002 and $50,000$50,000 in 2001.2001. Do the sample data indicate an increase in the mean loan amount between 2001 and 2002?2002? Use =.05=.05

Gasoline prices reached record high levels in 16 states during 2003 (The Wall Street Journal, March 7,2003 ). Two of the affected states were California and Florida. The American Automobile Association reported a sample mean price of $2.04$2.04 per gallon in California and a sample mean price of $1.72$1.72 per gallon in Florida. Use a sample size of 40 for the California data and a sample size of 35 for the Florida data. Assume that prior studies indicate a population standard deviation of .10 in California and .08 in Florida are reasonable. a. What is a point estimate of the difference between the population mean prices per gallon in California and Florida? b. At 95%95% confidence, what is the margin of error? c. What is the 95%95% confidence interval estimate of the difference between the population mean prices per gallon in the two states?