Download Homework5 Estimation Final

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NOTE: All questions are worth 1 point, unless otherwise noted.
Homework 5
Laurie James and Laura Zylstra
Reading Questions
In this module we will examine the different sampling methods. We will see how the method of sampling
impacts the estimates one obtains from a sample of data. In particular we will examine two properties that
are desirable: unbiasedness and precision.
Answer the following questions regarding the “readings”:
1. How does random sampling provide samples that are representative of the population they are drawn
from? Random sampling provides samples that are representative of the population by avoiding
problems caused by voluntary response bias and undercoverage bias. For example, if people volunteer
or self-select to participate in a survey—they may be more likely to have strong opinions which may
skew the survey results in one direction or the other. By contrast, randomly selecting—by chance—
ensures that everyone in the population under study has an equal opportunity to be a participant and to
voice their opinions in the survey. This makes a higher likelihood that every group is equally
represented.
2. What type of survey bias does random sampling NOT reduce? Random sampling does not protect
against response bias—problems in the measurement process (also called measurement error).
Examples of response bias found in surveys not reduced by random sampling include: Nonresponse
bias (when individuals selected for the survey are unwilling or unable to participate), leading
questions (the manner in which survey questions are asked—resulting in skewed responses), and
social desirability (when respondents offer “favorable” answers which put themselves in the “best
light”—particularly if the survey does not ensure confidentiality).
3. What type of survey bias can a convenience sample sometimes cause? A convenience sample (when
samples are selected via a method that is convenient for the researcher) can lead to a type of survey
bias referred to as undercoverage or underrepresentation of certain members of the population. For
example, asking survey questions of SPU students while they are eating in Gwinn Commons (the oncampus cafeteria), will cause undercoverage of the SPU students who commute to campus or are
enrolled in online classes.
4. How do you know when a sample statistic is unbiased? A sample statistic is unbiased when every
member of the population has an equal chance of selection. When the mean of the sample statistic
distribution is equal to the true value of the population. The confirmation that a sample statistic is
unbiased is when “the average of all possible samples equals the true population parameter” (2012,
Stat Trek, p. 3). You know your survey results are unbiased if the averaged results of repeated
surveys fall in line with the true population parameter.
5. Does increasing the sample size reduce sampling error? Explain. Increasing the sample size (selecting
more people to participate in a survey, for example) can reduce sampling error by creating a more
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“stable” or focused range of possibilities and thus—more accurate estimate. Adding more participants
reduces the standard error of the mean. It allows for broader representation and thus a more accurate
picture of the population.
6. Does increasing the sample size reduce survey bias? Explain. Increasing the sample size does not
reduce survey bias (which results from conducting a survey in an incorrect manner). For example,
surveying an even larger number of people regarding their opinions on a subject will still lead to
biased results found in a smaller sample if the researcher continues to use “leading questions” or
neglects to ensure confidentiality (methodological problems). Additionally, if a surveyor selects
participants systematically, not randomly, the survey will still be biased. Increasing the number of
participants will have no effect on the bias.
7. The standard error is a measure of the precision of the sample estimate. This shows the variability in
scores so that one may predict the actual population parameters.
8. How does sample size affect the margin of error? Yes, the size of the sample does affect the margin of
error based on the following logic: This week’s module displayed an example indicating that a sample
size of 10 had a standard error of the mean of .64, while a larger sample of 25 had a decreased
standard error of the mean of .43. We know that “the standard error of the mean” measures the
variability or how spread out the sample means are and that a smaller number (derived from the larger
sample) provides a more accurate estimate (a tighter or smaller range of possibilities). Since the
“margin of error” is most often based on the formula (2SE)—then two times a smaller number results
in a narrower margin. Therefore: A larger sample size-leads to a smaller standard error of the
meanleads to a narrower margin of errorleads to a more accurate measure or estimate.
Write a report that answers the research question “What is the average word length in the Gettysburg
address?” (4pts).
The report should include:
– how the interval was computed,
– what the interval is,
– an interpretation of the interval and
– appropriate conclusions based on the interval and study design.
–
In an inferential statistical study designed to estimate the average word length in the Gettysburg address,
using SRS (Simple Random Sampling (important for creating a reprentative sample). TinkerPlots was
used to draw 500 simple random samples of 25 words each from the total population of 268 contained
within the passage (see Table 1). The average word length for each sample was then collected and plotted
(again, using the TinkerPlots software) resulting in the average of 500 sample estimates (n=25) being 4.
The distribution of the sample estimates in this simulation resulted in a standard error of the mean, or
standard deviation, of .42. Using the formula for a 95% confidence interval (mean + 2SE), the following
calculations were made: Margin of error: 2x .42=84, Lower confidence level: 4.30- .84=3.46
Upper confidence level: 4.30+.84=5.14. The resulting confidence interval of 3.46 to 5.14, indicates that
there is a 95% chance that the average word length in the Gettysburg address is between 3.46 and 5.14
letters long. Based on these results we can draw the conclusion that the average word length is between
3.46 to 5.14 with a 95% confidence level.
9.
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Table 1. (From Module 9, Powerpoint, slide # 13, Dr. Mvududu, Seattle Pacific University, 2012)
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