# Econ 2500-Assignment 2-Let X be a continuous random variable

Econ 2500 Section D 3.0 – Fall 2016Assignment 2(Due Date: November 24, 2015)Question 1: Let X be a continuous random variable with probability density function 0<x?22<x?4else cxf (x) = c(4 ? x) 0 a. Find c such that f (x) is a probability density function.b. Find P (0 < X < 1).c. Find P (1 ? X ? 3).d. Find P (X > 2|1 ? X ? 3).Question 2:Statistics Canada reported that 13% of Canadians are left-handed. 12Canadians are randomly selected.a. Let X be the number of right-handed Canadians being selected. Find the probabilitydistribution of X.b. What is the probability that majority of the selected Candians are right-handed.c. What is the mean and standard deviation of X.Question 3: The weight of potato chips in a medium size bag is stated to be 300 g.The amount that the packaging machine puts in these bags is believed to have be normallydistributed with mean 306 g and standard deviation of 3.6 g.a. What proportion of all bags sold are underweight?b. What is the probability that the mean weight of the three bags is below the statedamount?c. What is the probability that the mean weight per bag of a 24-bag case of potato chipsis below 300 g?Question 4: An automatic character recognition device can successfully read about 85%of hand-written credit card application.a. Three credits card applications are randomly selected. What is the probability thatthe device successfully read at least two of them?1 b. Three hundred credits card applications are randomly selected. What is the probabilitythat the device successfully read at least two hundred of them? For this question, firstgiven the exact expression of the required probability. Then approximate it based onthe method you have learned in class (or in text).c. What theory did you apply to obtain the answer in part (b)? Explain by the theory isapplicable in this case.Question 5:each part. Clearly define the variables and parameters and set up the hypotheses for a. In 2010, Bell Canada took an average 0.4 hours to respond to trouble calls. Bell Canadaobtained the data for 2014. Is there any evidence that show a change in the averageresponse time?b. US Census Bureau data show that the mean household income in the area served bya shopping mall is $42800 per year. A market research firm questions shoppers at themall to find out whether the mean household income of mall shoppers is higher thanthat of the general population.c. The average square footage of one-bedroom apartments in a new student-housing development is advertised to be 880 square feet. A student group thinks that the apartmentsare smaller than advertised. They hire an engineer to measure a sample of apartmentsto test their suspicion.d. To study the sleeping pattern of adolescents, a sample of 50 adolecscents, all free ofsevere illness, are randomly selected. The research believe that people with low sleepefficiency had an average of systolic blood pressure to be higher than 132 mm Hg.Question 6: A nutrition lab tested the mean sodium content of 100 hot dogs. The labreported that the mean sodium content of their sample is 322 mg with a standard deviationof 2.5 mg.a. One of the aims is to report a 95% confidence interval for the mean sodium content ofhot dogs. Is it necessary to assume the distribution of sodium content of hot dogs isnormally distributed? Why?b. Based on your result in part (a), obtain the 95% confidence interval for the meansodium content of hot dogs.c. Based on the result in part (b), is there any evidence that the mean sodium content isdifferent from the stated 325 mg on the package?d. A dietitian is interested to know if the result suggested that the mean sodium contentof hot dogs is significantly lower than the stated 325 mg on the package. What can youtell from the lab results? (You answer has to be justified based on statistics concepts.)2 Question 7: Consumer Reports tested 14 brands of vanilla yogourt and found the followingnumbers of calories per serving:160130 200170 220190 230 120 180 14080 120 100 170 a. State all the necessary assumptions and report a 90% confidence interval for the meancalorie content of vanilla yogourt.b. A diet guide claims that you will get an average of 120 calories from a serving of vanillayogourt. Based on the given data, is there evidence that the mean calorie content isdifferent from the claim of the diet guide?Question 8: A medical researcher believes the body temperature of adults is normallydistributed with a standard deviation of 0.4o C.a. What sample size is required such that a 99% confidence interval of mean body temperature of adults has width at most 0.06o C?b. Based on the result in part (a), is it really necessary to assume the body temperatureof adults is normally distributed? Why? 3