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CHAPTER 9 – ALMOST AT INFERENTIAL!!!!
CHAPTER 9.1 – SAMPLING DISTRIBUTIONS
Population is who or what you are interested in finding out about. Parameters are all
the numerical summaries that represent the population.
Statistics (from samples) are unbiased estimators of the true population parameters.
CHAPTER 9.2 – SAMPLE PROPORTIONS
The mean of the sampling distribution of p̂ is  pˆ  p .
The standard deviation of the sampling distribution of p̂ is  pˆ 
RULE OF THUMB 1
Use the recipe for standard deviation of
large as the sample.
p(1  p)
n
p̂ only when the population is at least 10 times as
RULE OF THUMB 2
We will use the normal approximation to the sampling distribution of
p that satisfy np ≥ 10 and n(1-p) ≥10.
p̂ for values of n and
CHAPTER 9.3 SAMPLE MEANS
The mean of the sampling distribution of x is
x  
The standard deviation of the sampling distribution is
x 

n
CENTRAL LIMIT THEOREM
Draw an SRS of size n from any population whatsoever with mean µ and finite standard
deviation σ. When n is large, the sampling distribution of the sample mean x is close to
the normal distribution N (  ,

n
) with mean µ and standard deviation σ.
1. Syracuse, NY is ranked the top city with the most average snowfall a year. Snowfall is
normally distributed there with a mean of 115.6” per year and a standard deviation of
20”.
(a) During what percentage of years does Syracuse get more than 100” of snow?
(b) Less than how much snow falls in the “driest” 25% of all years?
(c) Jon Squeri is a student at Syracuse for the next 4 years. What’s the probability that
those four years average between 120” and 140” of snow?
2. A recent study found the lifetimes of cell phones are normally distributed with a mean of
24.3 months and a standard deviation of 2.6 months.
(a) What’s the probability that your cell phone lasts more than 28 months?
(b) Between 23 and 25 months?
(c) What cell phone life lasts the shortest 15%?
(d) If a company provides its 33 employees with a cell phone, find the probability that
the mean lifetime of these phones will be less than 2 years.
(e) Between 23 and 25 months?
(f) Only 25% of this average employee’s cell phone life will last more than how many
years?
(g) Find the sample size necessary to insure that we have a 95% chance that our sample
mean is within 0.25 months of the population mean.
3. Two out of 5 adults smokers acquired the habit by age 14. Find the following
probabilities if 20 random adult smokers are selected:
(a) Find the mean and standard deviation of the amount of smokers that acquired the
habit by age 14 from this random sample.
(b) What’s the probability that exactly half of them acquired the habit by age 14?
(c) What’s the probability that more than 15 of them acquired the habit by age 14?
(d) Say we select 400 random smokers now, what is the probability that 150 or fewer of
them acquired the habit by age 14? More than half of them?
4. College students often make up a substantial portion of the population of college cities
and towns. State College, PA, ranks first with 71% of its population made up of college
students. What is the probability that in a random sample of 150 people from State
College, more than 50 are NOT college students?
5. According to a recent survey, 42% of Americans get at most 6 hours of sleep each
night. If 25 people are selected, find the probability that less than 30% of them will get
at most 6 hours of sleep.
6. Based on past experiences, a bank believes that 7% of the people who receive loans will
not make payments on time. The bank has recently approved 200 loans. What’s the
probability that over 10% of these clients will not make timely payments?
7. When a truckload of apples arrives at a packing plant, a random sample of 150 is
selected and examined for bruises, discoloration, and other defects. The whole
truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that
in fact 8% of the apples on the truck do not meet the desired standard. What’s the
probability that the shipment will be accepted anyway?
8. Just before a referendum on a school budget, a local newspaper polls 400 voters in an
attempt to predict whether the budget will pass. Suppose that the budget actually has
the support of 52% of the voters. What’s the probability the newspaper’s sample will
lead them to predict defeat?
A summer resort rents rowboats to customers but does not allow more than four people
to the boat. Each boat is designed to hold no more than 800 pounds. Suppose the
distribution of adult males who rent boats, including their clothes and gear, is normally
distributed with a mean of 190 pounds and standard deviation of 10 pounds. If the
weights of individual passengers are independent, what is the probability that a group of
four adult male passengers will exceed the acceptable weight limit of 800 pounds?
(a) 0.023
(b) 0.046
(c) 0.159
(d) 0.317
(e) 0.977