**Answers to
Study Questions**

**for **

**Chapter 16**

**(Don’t forget that the companion website also has multiple choice
questions for each chapter that you can take for practice. You will find them
here: http://www.southalabama.edu/coe/bset/johnson/dr_johnson/2mcq.htm)**

**16.1. What is the difference between a statistic and a parameter?**

A __statistic__ is a
numerical characteristic of a sample, and a __parameter__ is a numerical
characteristic of a population.

**16.2. What is the symbol for the population mean?**

The symbol is the Greek
letter mu (i.e., µ).

**16.3. What is the symbol for the population correlation coefficient?**

The symbol is the Greek
letter rho (i.e., ρ ).

**16.4. What is the definition of a sampling distribution?**

The __sampling distribution__
is the theoretical probability distribution of the values of a statistic that
results when all possible random samples of a particular size are drawn from a
population.

**16.5. How does the idea of repeated sampling relate to the concept of a
sampling distribution?**

__Repeated sampling__ involves drawing many or
all possible samples from a population.

**16.6. Which of the two types of estimation do you like the most, and
why?**

This is an opinion question.

·
__Point estimation__ is nice because it provides an exact point estimate of the population
value. It provides you with the single best guess of the value of the
population parameter.

·
__Interval estimation__ is nice because it allows you to make statements of confidence that an
interval will include the true population value.

**16.7. What are the advantages of using interval
estimation rather than point estimation?**

The problem with using a
point estimate is that although it is the single best guess you can make about
the value of a population parameter, it is also usually wrong.

·
Take
a look at the sampling distribution of the mean on page 468 and note that in
that case if you would have guessed $50,000 as the correct value (and this WAS
the correct value in this case) you would be wrong most of the time.

·
A
major advantage of using interval estimation is that you provide a range of
values with a known probability of capturing the population parameter (e.g., if
you obtain from SPSS a 95% confidence interval you can claim to have 95%
confidence that it will include the true population parameter.

·
An
interval estimate (i.e., confidence intervals) also help one to not be so
confident that the population value is exactly equal to the point estimate.
That is, it makes us more careful in how we interpret our data and helps keep
us in proper perspective.

·
Actually,
perhaps the best thing of all to do is to provide both the point estimate and
the interval estimate. For example, our best estimate of the population mean is
the value $32,640 (the point estimate) and our 95% confidence interval is
$30,913.71 to $34,366.29.

·
By
the way, note that the bigger your sample size, the more narrow the confidence
interval will be.

·
If
you want narrow (i.e., very precise) confidence intervals, then remember to
include a lot of participants in your research study.

**16.8 What is a null hypothesis?**

A __null hypothesis__ is
a statement about a population parameter. It usually predicts no difference or
no relationship in the population. The null hypothesis is the “status quo,” the
“nothing new,” or the “business as usual” hypothesis. It is the hypothesis that
is directly tested in hypothesis testing.

**16.9. To whom is the researcher similar to in hypothesis testing: the
defense attorney or the prosecuting attorney? Why?**

The researcher is similar to
the *prosecuting attorney* is the sense that the researcher brings the
null hypothesis “to trial” when she believes there is probability strong
evidence against the null.

·
Just
as the prosecutor usually believes that the person on trial is not innocent,
the researcher usually believes that the null hypothesis is not true.

·
In
the court system the jury must assume (by law) that the person is innocent
until the evidence clearly calls this assumption into question; analogously, in
hypothesis testing the researcher must assume (in order to use hypothesis
testing) that the null hypothesis is true until the evidence calls this
assumption into question.

**16.10. What is the difference between a probability value and the significance
level?**

Basically in hypothesis
testing the goal is to see if the probability value is less than or equal to
the significance level (i.e., is p ≤ alpha).

·
The
__probability value__ (also called the p-value) is the probability of the
result found in your research study of occurring (or an even more extreme
result occurring), under the assumption that the null hypothesis is true.

·
That
is, you assume that the null hypothesis is true and then see how often your
finding would occur if this assumption were true.

·
The
__significance level__ (also called the alpha level) is the cutoff value the
researcher selects and then uses to decide when to reject the null hypothesis.

·
Most
researchers select the significance or alpha level of .05 to use in their
research; hence, they reject the null hypothesis when the p-value (which is
obtained from the computer printout) is less than or equal to .05.

**16.11. Why do educational researchers usually use .05 as their
significance level?**

It has become part of the
statistical hypothesis testing culture.

·
It
is a convention.

·
It
reflects a concern over making type I errors (i.e., wanting to avoid the
situation where you reject the null when it is true, that is, wanting to avoid
“false positive” errors).

·
If
you set the significance level at .05, then you will only reject a true null
hypothesis 5% or the time (i.e., you will only make a type I error 5% of the
time) in the long run.

** **

**16.12. State the two decision making rules of hypothesis testing.**

·
__Rule one__: __If__ the p-value is less than or equal to the significance level
__then__ reject the null hypothesis and conclude that the research finding
is statistically significant.

·
__Rule two__: __If__ the p-value is greater than the significance level __then__
you “fail to reject” the null hypothesis and conclude that the finding is not
statistically significant.

**16.13. Do the following statements sound like typical null or
alternative hypotheses? (A) The coin is fair. (B) There is no difference
between male and female incomes in the population. (C) There is no correlation
in the population. (D) The patient is not sick (i.e., is well). (E) The
defendant is innocent.**

All of these sound like null
alternative hypotheses (i.e., the “nothing new” or “status quo” hypothesis). We
usually assume that a coin is fair in games of chance; when testing the
difference between male and female incomes in hypothesis testing we assume the
null of no difference; when testing the statistical significance of a
correlation coefficient using hypothesis testing, we assume that the correlation
in the population is zero; in medical testing we assume the person does not
have the illness until the medical tests suggest otherwise; and in our system
of jurisprudence we assume that a defendant is innocent until the evidence
strongly suggests otherwise.

**16.14. What is a Type I error? What is a Type II error? How can you
minimize the risk of both of these types of errors?**

In hypothesis testing there
are two possible __errors__ we can make: Type I and Type II errors.

·
A
__Type I error__ occurs when your reject a true null hypothesis (remember
that when the null hypothesis is true you hope to retain it).

·
A
__Type II error__ occurs when you fail to reject a false null hypothesis
(remember that when the null hypothesis is false you hope to reject it).

·
The
best way to allow yourself to set a low alpha level (i.e., to have a small
chance of making a Type I error) and to have a good chance of rejecting the
null when it is false (i.e., to have a small chance of making a Type II error)
is to ** increase the sample size**.

·
The
key in hypothesis testing is to use a large sample in your research study
rather than a small sample!

·
If
you do reject your null hypothesis, then it is also essential that you
determine whether the size of the relationship is practically significant (see
the next question).

**16.15. If a finding is statistically significant, why is it also
important to consider practical significance?**

When your finding is __statistically
significant__ all you know is that your result would be unlikely if the null
hypothesis were true and that you therefore have decided to reject your null
hypothesis and to go with your alternative hypothesis. Unfortunately, this does
not tell you anything about how big of an effect is present or how important
the effect would be for practical purposes. That’s why once you determine that
a finding is statistically significant you must next use one of the __effect
size indicators__ to tell you how strong the relationship. Think about this
effect size and the nature of your variables (e.g., is the IV easily
manipulated in the real world? Will the amount of change relative to the costs
in bringing this about be reasonable?).

·
Once
you consider these additional issues beyond statistical significance, you will
be ready to make a decision about the __practical significance__ of your
study results.

**16.16. How do you write the null and alternative hypotheses for each of
the following: (A) The t-test for independent samples, (B) One-way
analysis of variance, (C) The t-test for correlation coefficients?, (D)
The t-test for a regression coefficient.**

In each of these, the null
hypothesis says there is no relationship and the alternative hypothesis says
that there is a relationship.

(A) In this
case the null hypothesis says that the two population means (i.e., mu

one and mu two) are equal; the alternative
hypothesis says that they are not

equal.

(B) In this
case the null hypothesis says that all of the population means are equal;

the alternative hypothesis says that at least
two of the means are not equal.

(C) In this
case the null hypothesis says that the population correlation (i.e., rho)

is zero; the alternative hypothesis says that
it is not equal to zero.

(D) In this
case the null hypothesis says that the population regression coefficient

(beta) is zero, and the alternative says that
it is not equal to zero.

You can examples of these
null and alternative hypotheses written out in symbolic form for cases A, B, C,
and D in the following Table.