1. Billy Bonka’s candy factories make dancing gummibears. Billy Bonka wants them all to be the same size (so that they can best dance together), but lately the Loompa-Oompas (Bonka’s
factory workers) have begun to make the dancing gummibears many different sizes. After
examining the latest batch, Billy Bonka found that the Loompa-Oompas are still making the
gummibears 6 centimeters tall on average, as he would want, but the standard deviation is
now 1.5 centimeters, which is larger than the .5 centimeters he prefers. Assuming that the
Loompa-Oompas are making all the dancing gummibears with heights following a normal
distribution with a mean of 6 centimeters and a standard deviation of 1.5 centimeters, answer
the following questions:
(a) Are the numbers 6 and 1.5 parameters or statistics? What symbols do we use to denote
them?
(b) Draw a sketch of the distribution of gummibear heights.
(c) What is the probability of a single dancing gummibear being more than 8 cm tall?
(d) What is the probability of a single dancing gummibear being less than 4.5 cm tall?
(e) What proportion of dancing gummibears are between 6 and 7 cm tall?
2. Billy Bonka’s whistling whoppers sing one of four different tunes when they are taken out
of the bag - “My Country ’tis of Thee,” “Let It Go,” “Yellow Submarine,” and “Livin’ on
a Prayer.” Assuming that 27% of the whistling whoppers made whistle “Let It Go,” answer
the following questions:
(a) Is the 27% a parameter or a statistic? What symbol do we use to denote it?
(b) If we take a sample of 40 whistling whoppers, is the Central Limit Theorem valid here?
(c) Suppose we take a sample of 40 whistling whoppers and calculate the sample proportion
of how many sing “Let It Go.” What does the Central Limit Theorem tell us about how
this sample proportion varies from sample to sample?
(d) Sketch the sampling distribution of the sample proportion.
(e) What is the probability that a the proportion of whistling whoppers that whistle ”Let
It Go” in a sample of 40 is greater than 33%?
3. Billy Bonka is beginning to suspect that the Loompa-Oompas have decreased the number of
whistling whoppers that whistle “Let It Go” (possibly because they are tired of the song).
He samples 120 whistling whoppers and finds that only 11 of them play “Let It Go.”
(a) What proportion of the 120 whistling whoppers play “Let It Go?” Is this a proportion
or statistic? What symbol do we use to denote this?
(b) Billy Bonka’s chief Loompa-Oompa assures Bonka that, while they are making fewer
“Let It Go” whistling whoppers, the proportion has only dropped to .2. We want to test
if the proportion has decreased past .2. Give the hypotheses for this significance test.
Be sure to define π.
(c) Is the central limit theorem valid to use here? Why or why not?
(d) Find the test statistic for this significance test.
(e) Find the p-value and interpret the p-value for this test.
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(f) Given a significance level of .05, what can you conclude?
4. Billy Bonka wants an estimate of what proportion of “Let It Go” whistling whoppers are
being made. As before, he samples 120 whistling whoppers and finds that only 11 of them
play “Let It Go.”
(a) Give a 90% confidence interval for the proportion of candies that play “Let it Go.”
(b) Interpret this interval.
(c) Explain what you mean when you say “90% confident.”
(d) What are two things we could do to decrease the margin-of-error for this confidence
interval?
(e) If he wanted to perform a two-sided significance test at the α = .10 level to tell if the
proportion is not equal to .15, what would he decide? Why?
(f) Could he use this confidence interval to predict the result of a significance test at α = .05
of the hypothesis that the proportion has decreased from .15? Why or why not?