Leader: Carly
Course: Stat 226
Date: 3/13/13
Confidence Intervals
Chapter 15
Supplemental Instruction
Iowa State University
Agenda
Opening Activities: Reviewing Terms
Main Activity: Practicing Confidence Intervals
Closing Activity: True or False Questions & Survey
Define the following terms in either words or with a formula:

Sampling Distribution: all possible values taken by the statistic in all possible samples of
the same size from the same population

Central Limit Theorem: the sampling distribution of the sample mean is approximately
normal with mean µ and standard error σ/sqrt(n) if it is drawn from a population with
mean µ and standard deviation σ with a sufficiently large sample size

Law of Large Numbers: as n increases, the sample mean gets closer to the population
mean in value

Confidence Interval: an interval of plausible values for µ constructed with a chosen
degree of confidence

Critical Value: 𝑧𝛼/2

Margin of Error: 𝑚 = 𝑧𝛼 ∗
2
𝜎
√𝑛
= half of the confidence interval’s width
Label each statement as true or false. For the statements that are false, change them in a
way that makes them correct.
For a sufficiently large sample size, we can typically expect the observed sample mean to be
fairly close in value to the population mean. True
The 68-95-99.7 Rule says that approximately 95% of all times we estimate the sample mean
(from some sufficiently large sample size) from a population with an unknown mean and a
known standard deviation we will produce sample means that fall within 2 standard deviations of
the population mean. True
“C” can be interpreted as the short-run probability of success for the method used to construct
CI. False; long-run
The desired level of confidence C determines which critical value to use. True
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A random sample of 30 students are selected from an accounting masters program and
information about the average number of hours per week they spend preparing for their
CPA examination is collected. The sample yielded an average of 22 hours per week.
Assume the population distribution is normal and the standard deviation is known to be 2
hours. Answer the following questions:
What is the sample size? 30
What is the sample mean? 22
What is the standard error of the sampling distribution? 0.37
Calculate a 90% confidence interval for the unknown population mean. Provide an
interpretation in the context of this problem.
We can be 90% confident that the true population average hours spent studying for the
CPA per week among accounting masters program is between 21.40 and 22.60 hours/week.
Calculate a 95% confidence interval for the unknown population mean. Provide an
interpretation in the context of this problem?
We can be 95% confident that the true population average… is between 21.28 and 22.71
hours per week.
Why are the interpretations of these confidence intervals in terms of the unknown
population mean and not the sample mean?
Because we already know that the sample mean falls exactly in the center of the confidence
interval. We are trying to make statements about the true population mean, since it is
unknown.
Which of the two confidence intervals is wider? Is this what you would expect?
The 95% confidence interval is wider. This is what we would expect because, as we
increase confidence, we sacrifice precision, which means the interval is wider.
What would happen to the 90% confidence interval if we increased the sampling size to 100
masters of accounting students?
The 90% confidence would get narrower. It would now have a width of 0.658, as opposed
to 1.2014 calculated from a sample size of 30.
What are the necessary assumptions for constructing this confidence interval?
Although we don’t have to “assume” these are true here, the assumptions we would
typically use for constructing confidence intervals are that 1) the sample mean is normally
distributed and 2) the sample is randomly selected.
A random sample of 49 business managers within a nation-wide banking corporation
collects the number of hours spent in managerial meetings per month. The population
mean is unknown, but the standard deviation of the population is known. If we construct a
99% confidence interval and obtain an interval of (38.528, 41.472) hours per month, what
is the sample mean? What is the standard deviation of the population?
Sample mean = 40; standard deviation = 4