Download 1-Chapter 9 - Vocab

CHAPTER 9 –HYPOTHESIS TESTING
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.
Chapters 8 and 9 – Confidence Intervals and Hypothesis testing for one population
(I) Sections 8.3 and 9.3 - Confidence Intervals and Hypothesis testing Regarding the Population Mean μ (σ Known/Unknown)
Assumptions
 We have a simple random sample
 The population is normally distributed or the sample size, n, is large (n > 30)
o
o
o
The procedure is robust, which means that minor departures from normality will not adversely affect the results of
the test. However, for small samples, if the data have outliers, the procedure should not be used.
Use normal probability plots to assess normality and box plots to check for outliers.
A normal probability plot plots observed data versus normal scores. If the normal probability plot is roughly linear
and all the data lie within the bounds provided by the software (our calculator does not show the bounds), then we
have reason to believe the data come from a population that is approximately normal.
(II) Using the calculator to test hypothesis or construct confidence intervals for one population mean


For hypothesis testing use items 1 or 2 from the STAT, TESTS menu
For confidence intervals use items 7 or 8 from the STAT, TESTS menu
o
Notice: The Z-procedures are used when sigma, the population standard deviation is given. If you only have
access to s, the standard deviation of a sample, use the T-procedures
(III) Sections 8.2 and 9.2 - Confidence Intervals and Hypothesis Testing Regarding the Population Proportion
Assumptions
 The sample is a simple random sample. (SRS)
 The conditions for a binomial distribution are satisfied by the sample. That is: there are a fixed number of trials, the trials
are independent, there are two categories of outcomes, and the probabilities remain constant for each trial. A “trial”
would be the examination of each sample element to see which of the two possibilities it is.
 The normal distribution can be used to approximate the distribution of sample proportions when
np ≥ 10 and n(1 – p) ≥ 10 are both satisfied. For confidence intervals use np ≥ 15 and n(1 – p) ≥ 15
o
Technically, many times the trials are not independent, but they can be treated as if they were independent if n
≤ 0.05 N (the sample size is no more than 5% of the population size)
o
Notice that it is possible that in some cases the p-value method may yield a different conclusion than the
confidence interval method. This is due to the fact that when constructing confidence intervals, we use an
estimated standard deviation based on the sample proportion p-hat.
o
If we are testing claims about proportions, it is recommended to use the p-value method or the traditional
method.
(IV) Using the calculator to test hypothesis or construct confidence intervals for one population proportion


For hypothesis testing use item 5 from the STAT, TESTS menu
For confidence intervals use item A from the STAT, TESTS menu
VOCABULARY

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).

In practice, we do not know σ. Replace the standard deviation
of
by the standard error
get the one-sample t statistic
The t statistic has the t distribution with n − 1 degrees of freedom.

When dealing with the t-distribution, p-values will be obtained with the calculator
to