Assignment 2. Due Wednesday 4:30pm, June 6 ion the dropbox. (Hard set because I want to send out
the keys immediately after).
This assignment is designed to be a warm up for the midterm. It should take 4-6 hours, assuming the
last one took 6-9 hours. I know a week is a short time but I want to give a cushion for Friday to handle
any confusion.
Problem 1) (Based on real data, real story).
When nutrition guides were being made, a large group of healthy people’s diets were surveyed. The
intake of many nutrients was found to be normally distributed. The amounts that were two
standard deviations below the mean were set as the recommended daily minimums.
a) What proportion of surveyed healthy people met or exceeded this minimum for calcium?
b) Consider three surveyed healthy people. What is the probability that all three are getting
enough calcium?
c) Adult men in the survey took in 14 mg/day of zinc, with a standard deviation of 1.5 mg/day,
what is the daily recommended minimum of zinc for adult men?
d) All adult men, healthy or otherwise, take a mean of 12mg/day of zinc and a standard deviation
of 2 mg/day. What proportion of them get their daily recommended amount of zinc?
Problem 2) (Real data)
According to a recent UN report, 2.5 million of Canada’s 34 million citizens are food insecure, meaning
that continual access to healthful food is not guaranteed for these people.
a) What is the probability that a randomly selected Canadian is food insecure?
b) Consider two randomly selected people. Find the probability that both are food insecure.
c) Consider two randomly selected people. Find that probability that one is food insecure but the
other is not.
d) Consider a family with two children. Find the probability that both children are food insecure or
explain why you can’t with only this information.
Problem 3) Consider the standard score / raw score formulae:
a) There are some kinds of values that sigma can never take, what are they?
b) If a z-score is positive, which is larger: the raw score x, or the mean? (Justify, no marks for
guessing)
Problem 4) (Modified from an old textbook question, but based on real data)
IQ, or intelligence quotient, used to be considered a good measure of general intelligence. IQ scores
were found to be normally distributed with a mean of 100 and standard deviation of 15. The questions
are updated regularly, one reason is that people as a whole are getting better at the test such that
someone taking the test in 2012 would score 20 points higher on the 2012 test than they would on the
1950 test.
So someone scoring a 93 today would get 113 on the 1950 test.
a) Anyone scoring over 130 is considered a genius. What proportion of today’s people would be
considered geniuses on today’s test?
b) If many people today took the 1950 test, what would be their average score?
c) What proportion of people today would be considered geniuses by the 1950 test? (Assume the
standard deviation is still 15).
(For interest: Are people getting smarter or just better at taking tests? You would know better than I.)
Problem 5) Which are statistics and which are parameters? Identify the population of interest for every
question and identify the sample when applicable. (Sources are for interest only, everything you need is
in the sentence.)
a) “The federal agency said the census data showed that 85.2 per cent of Canada's population was
under the age of 65. “(Source: Canada.com, “Canada still youngest amongst G8” May 29,2012)
b) 25 courses were randomly selected from all those available in the summer, and an average of
2.4 required textbooks was found. (modified from your textbook).
c) A research organization conducted a survey and found that 45% of respondents felt they were
better off economically than before.
d) “Statistics Canada says the caseload in youth courts fell seven per cent in 2010-2011, a second
straight annual drop. The agency says youth courts handled about 52,900 cases last year.”
(source: Toronto Star: Youth Courts Caseload Falling: Statscan.” May 28, 2012)
e)
“A survey featured in [a recent] report reveals 82 per cent of mothers cite safety concerns as
reasons why they restrict outdoor play [of their children].” (Source: The Globe and Mail: “Canadian
kids get failing grade in physical activity: report.” May 29, 2012)
Problem 6) Which of the following sampling methods are being used in each situation? The methods
are: Accidental Sampling, Simple Random Sampling (SRS), and Stratified Sampling. (There are plenty of
other methods but these are three we care about.)
a) A truckload of oranges in crates are tested for rot by selecting a few crates from the truck and
selecting a few oranges from each crate.
b) A news site has an opinion poll on their front page about a local issue.
c) A pollster calls 1,000 random telephone numbers with the 604 or 778 extension to get opinions
in Vancouver.
d)
(bonus) A pollster calls 1,000 random people in the Vancouver phone book to get opinions.
e) The university sends compulsory surveys to 500 of the 5,000 grad students and 2,000 of the
20,000 undergrad students (compulsory means required).
f)
10,000 surveys are sent out to randomly selected people, and those people are required by law
to complete them.
g) 15,000 surveys are sent out to randomly selected people, but those people volunteer to
complete them or not.
Problem 7) Hummus, a food product, comes in packages that claim there is 227 grams of hummus inside
the container. In reality not every container has the exactly the same amount of hummus. The weight of
the product has a normal distribution with mean 232 grams, and standard deviation of 4 grams.
a) What is the probability that a randomly selected hummus package is underweight (contains less
than the advertised amount)?
b) What is the probability that an average of 4 packages is less than 227 grams?
c) What is the probability that an average of 25 packages is less than 227 grams?
d) Give an upper bound to the chance that an average of 400 packages is less than 227 grams?
e) If you took the average of a larger and larger sample, what would expect to happen to the
sample mean of the weights?
Problem 8) Suppose you are a vet and you’re inspecting a sample of 10 bearded dragons. You know that
dragons’ lengths are normally distributed and have a standard deviation of 5 cm.
a) What is the difference between standard deviation and standard error of the mean?
b) What is the standard error of the mean of your sample?
c) How many dragons would you need in your sample to a half of its old value? To a third?