Practice
Problems
1. Three things influence the margin of error in a confidence
interval estimate of a population mean: sample size, variability in the
population, and confidence level. For each of these quantities
separately, explain briefly what happens to the margin of error as that
quantity increases.
2. . A survey of 1000 Californians finds reports that 48% are excited
by the opportunity to take a statistics class.
Construct a 95% confidence interval on the true proportion of
Californians who are excited to take a statistics class. Be sure to
state/check assumptions.
3. Since your interval contains values above 50% and
therefore does find that it is plausible that more than half of the
state feels this way, there remains a big question mark in your mind.
Suppose you decide that you want to refine your estimate of the
population proportion and cut the width of your interval in half. Will
doubling your sample size do this? How large a sample will be needed to
cut your interval width in half? How large a sample will be needed to
shrink your interval to the point where 50% will not be included in a
95% confidence interval centered at the .48 point estimate?
4. A random sample of 67 lab rats are enticed to run through a maze,
and a 95% confidence interval is constructed of the mean time it takes
rats to do it. It is [2.3min, 3.1 min]. Which of the following
statements is/are true? (More than one statement may be correct.)
(A) 95% of the lab rats in the sample ran the maze in between 2.3 and
3.1 minutes.
(B) 95% of the lab rats in the population would run the maze in between
2.3 and 3.1 minutes.
(C) There is a 95% probability that the sample mean time is between 2.3
and 3.1 minutes.
(D) There is a 95% probability that the population mean lies between
2.3 and 3.1 minutes.
(E) If I were to take many random samples of 67 lab rats and take
sample means of mazerunning times, about 95% of the time, the sample
mean would be between 2.3 and 3.1 minutes.
(F) If I were to take many random samples of 67 lab rats and construct
confidence intervals of mazerunning time, about 95% of the time, the
interval would contain the population mean. [2.3, 3.1] is the one such
possible interval that I computed from the random sample I actually
observed.
(G) [2.3, 3.1] is the set of possible values of the population mean
mazerunning time that are consistent with the observed data, where
“consistent” means that the observed sample mean falls in the middle
(“typical”) 95% of the sampling distribution for that parameter value.
5. Two students are doing a statistics project in which they drop toy
parachuting soldiers off a building and try to get them to land in a
hulahoop target. They count the number of soldiers that succeed and
the number of drops total. In a report analyzing their data, they write
the following:
“We constructed a 95% confidence interval estimate of the proportion of
jumps in which the soldier landed in the target, and we got [0.50,
0.81]. We can be 95% confident that the soldiers landed in the target
between 50% and 81% of the time. Because the army desires an estimate
with greater precision than this (a narrower confidence interval) we
would like to repeat the study with a larger sample size, or repeat our
calculations with a higher confidence level.”
How many errors can you spot in the above paragraph?
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