CHAPTER 14 –INTRODUCTION TO INFERENCE
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Any topic listed on this document and not covered in class must be studied “On Your Own” (OYO)
READ INTRODUCTION
 What is statistical inference?
 Conditions for inference about a mean
Section 14.1 – THE REASONING OF STATISTICAL ESTIMATION (pg. 360)
Section 14.2 – MARGIN OF ERROR AND CONFIDENCE LEVEL (pg. 362
 Estimating the population mean
o Confidence interval
o Confidence level
o Margin of error
 Interpretation of a confidence interval
 Run the applet from page 364
Section 14.3 – CONFIDENCE INTERVALS FOR A POPULATION MEAN (pg. 364)
 Critical values for
o 90%, 95%, 99% confidence intervals
 Confidence interval for the mean of a normal population
Section 14.4 – THE REASONING OF TESTS OF SIGNIFICANCE (pg. 368)
 Distribution of sample means for samples of size n
o Simulation
 Population distributions
 Sampling distributions of a statistic (in this case the sample mean)
UNDERSTANDING HYPOTHESIS TESTING


The researcher uses data from a sample to evaluate the credibility of the null hypothesis.
The data will either provide support for Ho or tend to reject Ho.

We analyze the distribution of sample means, for all samples of size n to determine what sample means are
consistent with Ho and what sample means are at odds with Ho. This distribution has a mean of mu and a
standard error of sigma/sq.rt(n)
 This distribution is divided into two sections:
1. Sample means that are likely to be obtained if Ho is true, that is, sample means that are close to Ho
2. Sample means that are very unlikely to be obtained is Ho is true, that is, sample means that are very different
from the null hypothesis.

The high probability samples are located in the center of the graph; the low probability samples are located in
the extreme tails.

The extremely unlikely values of sample means make up the critical region or rejection region.

To find the boundaries of the two regions we select a probability value known as level of significance or the
alpha level

If alpha = 0.05, we are taking a 5% risk of committing a type I error (rejecting the null hypothesis when it is in
fact true)

The p-value P is the probability that the result occurred by chance alone.

Reporting the results
o In a scientific journal you will not be told explicitly that the null hypothesis has been rejected.
o You will see the statement:
 Results are statistically significant. The treatment with medication had a significant effect on
people’s depression scores. Z = 2.45, p = 0.0071. (this means the obtained result is not likely to
occur by chance)
 Test results are not statistically significant. There was no evidence that the medication had an
effect on depression scores, z = 1.30, p = .0968 (this means the obtained result is relatively likely
to occur by chance)

A significant result means that the hypothesis test has ruled out chance (sampling error) as a plausible
explanation for the results.
BOOK’S SUMMARY

A confidence interval uses sample data to estimate an unknown population parameter with an indication
of how accurate the estimate is and of how confident we are that the result is correct.

Any confidence interval has two parts: an interval calculated from the data and a confidence level C. The
interval often has the form
estimate ± margin of error

The confidence level is the success rate of the method that produces the interval. That is, C is the
probability that the method will give a correct answer. If you use 95% confidence intervals often, in the long
run 95% of your intervals will contain the true parameter value. You do not know whether or not a 95%
confidence interval calculated from a particular set of data contains the true parameter value.

A level C confidence interval for the mean µ of a Normal population with known standard deviation σ,
based on an SRS of size n, is given by

The critical value z* is chosen so that the standard Normal curve has area C between -z* and z*.

A test of significance assesses the evidence provided by data against a null hypothesis H0 in favor of an
alternative hypothesis Hα.

Hypotheses are always stated in terms of population parameters. Usually H0 is a statement that no effect is
present, and Ha says that a parameter differs from its null value in a specific direction (one-sided alternative)
or in either direction (two-sided alternative).

The essential reasoning of a significance test is as follows. Suppose for the sake of argument that the null
hypothesis is true. If we repeated our data production many times, would we often get data as inconsistent with
H0 as the data we actually have? Data that would rarely occur if H0 were true provide evidence against H0.

A test is based on a test statistic that measures how far the sample outcome is from the value stated by H0.

The P-value of a test is the probability, computed supposing H0 to be true, that the test statistic will take
a value at least as extreme as that actually observed. Small P-values indicate strong evidence against H0. To
calculate a P-value we must know the sampling distribution of the test statistic when H0 is true.


If the P-value is as small or smaller than a specified value a, the data are statistically significant at
significance level α.
Significance tests for the null hypothesis H0: µ = µ0 concerning the unknown mean µ of a population
are based on the one-sample z test statistic

The z test assumes an SRS of size n from a Normal population with known population standard deviation
σ. P-values can be obtained either with computations from the standard Normal distribution or by using
technology (applet or software).