Workshop 3
Section 3.1: Sampling Distributions
Example 1: Using Search Engines on the Internet
A 2012 survey of a random sample of 2253 US adults found that 1,329 of them reported using a search engine
(such as Google) every day to find information on the Internet.
a). Find the relevant proportion and give the correct notation with it.
b). Is your answer to part (a) a parameter or a statistic?
c). Give notation for and define the population parameter that we estimate using the result of part (a).
Example 2: Number of Books Read in a Year
A survey of 2,986 Americans ages 16 and older found that 80% of them read at least one book in the last year.
Of these book readers, the mean number of books read in the last year is 17 while the median number of books
read in the last year is 8.
a). How many “book readers” (defined as reading at least one book in the past year) were included in
the sample?
b). Why might the mean and median be so different? Using the information given about the mean and
median number of books read in a year, what is the likely shape of the distribution of number of books read in a
year by book readers?
c). Give the correct notation for the value “17” in the information above. Is this value a parameter or a
statistic?
d). Give notation for and define the population parameter that we estimate using the result of part (c).
Quick Self-Quiz: Parameters and Statistics
For each of the following, state whether the quantity described is a parameter or a statistic, and give the correct
notation.
a). The proportion of all residents in a county who voted in the last presidential election.
b). The mean number of extracurricular activities from a random sample of 50 students at your school.
c). The mean grade assigned for all grades given out to undergraduates at your school.
d). The difference in proportion who have ever smoked cigarettes, between a sample of 500 people
who are 60 years old and a sample of 200 people who are 25 years old.
Example 3: Proportion Never Married
A sampling distribution is shown for the proportion of US citizens over 15 years old who have never been
married, using the data from the 2010 US Census and random samples of size n = 500.
a). What does one dot in the dotplot represent?
b). Use the sampling distribution to estimate the proportion of all US citizens over 15 years old who
have never been married. Give correct notation for your answer.
c). If we take a random sample of 500 US citizens over 15 years old and compute the proportion of the
sample who have never been married, indicate how likely it is that we will see that result for each sample
proportion below.
̂ = 0.30 ̂ = 0.20 ̂ = 0.37 ̂ = 0.74
d). Estimate the standard error of the sampling distribution.
e). If we took samples of size 1000 instead of 500, and used the sample proportions to estimate the
population proportion:
Would the estimates be more accurate or less accurate?
Would the standard error be larger or smaller?
Quick Self-Quiz: Effect of Sample Size
Three different sampling distributions A, B, and C are given for a population with mean 50. One corresponds to
samples of size n = 25, one to samples of size n = 100, and one to samples of size n = 400. Match the sampling
distributions with the three sample sizes, and estimate the standard error for each.
Section 3.2: Understanding and Interpreting Confidence Intervals
Example 1: Adopting a Child in the US
A survey of 1,000 American adults conducted in January 2013 stated that “44% say it’s too hard to adopt a child
in the US.” The survey goes on to say that “The margin of sampling error is +/- 3 percentage points with a 95%
level of confidence.”
a). What is the relevant sample statistic? Give appropriate notation and the value of the statistic.
b). What population parameter are we estimating with this sample statistic?
c). Use the margin of error to give a confidence interval for the estimate.
d). Is 0.42 a plausible value of the population proportion? Is 0.50 a plausible value?
Example 2: Budgets of Hollywood Movies
A sampling distribution is shown for budgets (in millions of dollars) of all movies to come out of Hollywood in
2011, using samples of size n = 20. We see that the standard error is about 10.23. Find the following sample
means in the distribution and use the standard error 10.23 to find the 95% confidence interval given by each of
the sample means listed. Indicate which of the confidence intervals successfully capture the true population
mean of 53.48 million dollars.
̅ = 40
̅ = 70
̅ = 84
Quick Self-Quiz: Constructing Confidence Intervals
For each of the following, use the information to construct a 95% confidence interval and give notation for the
quantity being estimated.
a). ̂ = 0.72 with standard error 0.04
b). ̅ = 27 with standard error 3.2.
c). ̂1 − ̂2 = 0.05 with margin of error for 95% confidence of 0.02.
Quick Self-Quiz: Interpreting a Confidence Interval
Using a sample of 24 deliveries described in “Diary of a Pizza Girl” on the Slice website, we find a 95% confidence
interval for the mean tip given for a pizza delivery to be $2.18 to $3.90. Which of the following is a correct
interpretation of this interval? Indicate all that are correct interpretations.
a). I am 95% sure that all pizza delivery tips will be between $2.18 and $3.90.
b). 95% of all pizza delivery tips will be between $2.18 and $3.90.
c). I am 95% sure that the mean pizza delivery tip for this sample will be between $2.18 and $3.90.
d). I am 95% sure that the mean tip for all pizza deliveries in this area will be between $2.18 and $3.90.
e). I am 95% sure that the confidence interval for the mean pizza delivery tip will be between $2.18 and
$3.90.
Section 3.3: Constructing Bootstrap
Confidence Intervals
Example 1: Textbook Prices
Prices of a random sample of 10 textbooks (rounded to the nearest dollar) are shown:
$132 $87 $185 $52 $23 $147 $125 $93 $85 $72
a). What is the sample mean?
b). Describe carefully how we could use cards to create one bootstrap statistic from this sample. Be
specific.
c). Where will be bootstrap distribution be centered? What shape do we expect it to have?
Example 2: Reese’s Pieces
We wish to estimate the proportion of Reese’s Pieces that are orange, and we have one package of Reese’s
Pieces containing 55 pieces. Describe carefully how we can use this one sample to create a bootstrap statistic.
Be specific.
Quick Self-Quiz: Bootstrap Samples
A sample consists of the following values: 8, 4, 11, 3, 7.
Which of the following are possible bootstrap samples from this sample?
a) 8, 3, 7, 11
b). 4, 11, 4, 3, 3
c). 3, 4, 5, 7, 8
d). 7, 8, 8, 3, 4