Econ 2500 Section D 3.0 - Fall 2016 Assignment 2 | Complete Solution

Econ 2500 Section D 3.0 - Fall 2016 Assignment 2

Question 1: Let X be a continuous random variable with probability density function
f(x) =

cx 0 < x ≤ 2
c(4 − x) 2 < x ≤ 4
0 else
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. 12 Canadians are randomly selected.
a. Let X be the number of right-handed Canadians being selected. Find the probability distribution 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 normally distributed 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 stated amount?
c. What is the probability that the mean weight per bag of a 24-bag case of potato chips is 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 that the device successfully read at least two of them?
b. Three hundred credits card applications are randomly selected. What is the probability that the device successfully read at least two hundred of them? For this question, first given the exact expression of the required probability. Then approximate it based on the 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 is
applicable in this case.
Question 5: Clearly define the variables and parameters and set up the hypotheses for each part.
a. In 2010, Bell Canada took an average 0.4 hours to respond to trouble calls. Bell Canada obtained the data for 2014. Is there any evidence that show a change in the average response time?
b. US Census Bureau data show that the mean household income in the area served by a shopping mall is $42800 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher than that 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 apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion.
d. To study the sleeping pattern of adolescents, a sample of 50 adolecscents, all free of severe illness, are randomly selected. The research believe that people with low sleep efficiency 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 lab reported that the mean sodium content of their sample is 322 mg with a standard deviation of 2.5 mg.
a. One of the aims is to report a 95% confidence interval for the mean sodium content of hot dogs. Is it necessary to assume the distribution of sodium content of hot dogs is normally distributed? Why?
b. Based on your result in part (a), obtain the 95% confidence interval for the mean sodium content of hot dogs.
c. Based on the result in part (b), is there any evidence that the mean sodium content is different from the stated 325 mg on the package?
d. A dietitian is interested to know if the result suggested that the mean sodium content of hot dogs is significantly lower than the stated 325 mg on the package. What can you tell from the lab results? (You answer has to be justified based on statistics concepts.)
Question 7: Consumer Reports tested 14 brands of vanilla yogourt and found the following numbers of calories per serving:
160 200 220 230 120 180 140
130 170 190 80 120 100 170
a. State all the necessary assumptions and report a 90% confidence interval for the mean calorie content of vanilla yogourt.
b. A diet guide claims that you will get an average of 120 calories from a serving of vanilla yogourt. Based on the given data, is there evidence that the mean calorie content is different from the claim of the diet guide?
Question 8: A medical researcher believes the body temperature of adults is normally distributed with a standard deviation of 0.4 oC.
a. What sample size is required such that a 99% confidence interval of mean body temperature
of adults has width at most 0.06oC?
b. Based on the result in part (a), is it really necessary to assume the body temperature
of adults is normally distributed? Why?