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Sampling Distributions and Inferential Statistics Chapter 3
3.1-
Why are normal curves important?
Normal curves display a distribution of outcomes that appears in many samples.
Based on the normal curve below, what is the likelihood of a randomly selected person being
70-72 inches tall?
34 %
3.2-
Probabilities are reported as decimal proportions in statistics, but they can be converted to
percentages. How would you report 5% as a probability (a decimal proportion)? .05
3.3-
What is a z score, conceptually?
A z score is a measure of how far from the mean a raw score or observation falls.
Why are some z scores positive values while others are negative?
Because z scores measures distance from the mean, a z score can either be positive (
above the mean) or negative (below the mean )
3.4-
What is the total percentage of scores that lie to the right of the z score of +1.96? Use the z
score table in the appendix to find the answer.
0.025
3.5-
Given M = 14 and s = 4, what is the z score of a raw score of 11? -
0.75
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3.6-
Use formula 3.1b and the z table to solve this problem.
If a distribution has a mean of 130 and a standard deviation of 10, what is the probability of
randomly selecting a score above 140?
0.1587
When M = 34 and s = 3, what percentage of scores are lower than 28?
2.28
3.7-
True or false: Any distribution that is transformed into a Z distribution will become normal.
False
3.8-
What do inferential statistics allow researchers to do?
Draw conclusions about populations based on sample data.
3.9-
A dealer draws a card from one deck and then draws a card from another deck. A researcher
states that the likelihood of drawing a spade both times is one out of 16 or 6.25 percent. Which
of the following assumptions is the best example of “willful ignorance”? Both decks contain all
52 cards.
3.10-
What is the probability of selecting a spade from a deck of 52 cards?
0.25
3.12-
Why is random sampling so important in inferential statistics?
Random sampling maximizes the likelihood that a sample is representative of the
population.
A researcher studies a random sample of U.S. college students and finds that the average
student loan debt is $30,270. Why would it be inappropriate to use this figure to make
inferences about college student debt in Europe?
European college students were not part of the population the researcher studied.
3.13-
Why do researchers use hypothesis testing?
To establish the credibility of a given hypothesis about a population.
3.14-
What is a statistical hypothesis?
A numerical statement about the outcome of a study.
What is a null hypothesis?
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A hypothesis that states that there is no effect of the independent variable on the
dependent variable.
Suppose a local promoter, wanting to create a unique selling feature for their community,
decides to try to create larger squirrels by making and spreading genetically modified nuts
throughout the community that have been supplemented with a growth hormone. What would
the research hypothesis be for this experiment?
The research hypothesis would be that squirrels that eat the genetically modified nuts
will grow to become larger squirrels.
3.15-
If a statistical analysis suggests the null hypothesis should be rejected, this only means that the
alternative hypothesis is most likely true. Why is this the case?
For any inferences made in statistical analysis, researchers have to account for the
probabilistic nature of that conclusion.
3.16-
How would one conduct a theoretical sampling distribution of means?
By choosing a sample size, then taking every possible sample of that size from the
population and measuring each for a particular parameter.
3.17-
According to the central limit theorem, when could a sampling distribution NOT be normal? A
sampling distribution could be nonnormal when the raw population scores are wildly
nonnormal and the selected sample size is small.
How does the mean of a sampling distribution (of means) compare to the population mean of
the sampled population? They are equal.
3.18-
How is the variability of a sampling distribution affected by the sample size?
The variability of a sampling distribution decreases as the sample size increases.
Why is there not just one sampling distribution for a given population?
There are as many sampling distributions as there are sample sizes.
The height of a population of high school students from a small city is collected and the mean is
found to be 67 inches (5’7). In creating the sampling distribution, which sample size (n) is most
likely to produce a sample with a mean of 75 inches (6’3) ? n = 4
3.19-
What is a sampling error?
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The differences between a population parameter and the estimate of that parameter
provided by a statistic.
3.21-
Why should a researcher use z test instead of a t test?
When the population standard deviation is known.
What do z test and t test help the researcher decide?
If a sample mean likely comes from a specified population.
3.22-
Why are z scores useful to researchers?
Z scores measure how far a score is from the population parameter by a standard
measure.
What is the z statistic?
The z statistic transforms means within a sampling distribution into z scores.
3.23-
Z test involves transforming a sample’s mean into a z statistic. Why is this helpful to the
researcher?
The z statistic shows how unlikely selecting that sample would be assuming the null
hypothesis is true.
What does it mean that a hypothesis test is a test of a theoretical population?
When hypothesis testing, the researcher is setting up a theoretical population,
which is different from the existing population, and seeing if that difference has
an effect on a specific statistic.
3.24-
Choose the best answer from the options below and fill in the blank: The farther a z score
statistic is from 0,_____. the more likely it is that the null hypothesis should be
rejected.
3.25-
For a hypothesis test, a researcher decides to use an alpha level of .10. What does this
mean?
The researcher will reject the null hypothesis if the observed sample statistic is
less than or equal to 10% likely (p ≤ .10) to occur if the null hypothesis is assumed to be
true.
An alpha level of .025 corresponds to the critical values of -1.96 and +1.96. Why are
these critical values important?
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Z statistics above +1.96 and below -1.96 belong to samples that are less than .025
= 2.5% likely, assuming the null hypothesis is true.
3.26-
What is statistical significance?
A hypothesis test’s findings are statistically significant when they suggest that
the researcher should reject the null hypothesis.
A researcherv hypothesize that a driving course is effective at preventing vehicle accidents and
decides to test their hypothesis with a z test. The researcher takes a sample…
The researcher should reject the null hypothesis that the theoretical population of
all drivers who receive this intervention have the same number of accidents as
existing drivers.
3.27-
When should a single sample t test be used instead of a z test?
T tests should be used when the population standard deviation is unknown.
Why is Cohen’s d useful to experimental researchers?
Cohen’s d provides a standard measure for effect size.
3.28-
Degrees of freedom are always equal to the number of things that are “free” to vary,
unless there is a restriction on those things. Why is this the case?
If there is a restriction, then after selecting a number of things, one will know what
the remaining things are.
3.29-
Which of the following are three important aspects of the outcome of any inferential
test?
An outcome of an inferential test is evidence that is statistical and suggestive of a
conclusion.
3.30-
Just as in other methods, single-sample z and t test assume that any samples a
researcher gathers have which property? Representativeness
