Hypothesis Testing:

1) believe that the relationship we found in our sample data is the same as the relationship we would find if we tested the entire population

2) believe that the relationship we found in our sample data is a coincidence produced by sampling error

OR

These will help us to decide if we should:

Inferential statistics

Inferential statistics-Univariate statistical analysis: tests hypotheses involving only one variable

-Bivariate statistical analysis: tests hypotheses involving two variables

-Multivariate statistical analysis: tests hypotheses and models involving multiple variables

Specific Steps in Hypothesis Testing

1. Specify the hypothesis.2. Select an appropriate statistical test.3. Two-tailed or One-tailed test.4. Specify a decision rule (alpha = ???).

Compute the critical value.5. Calculate the value of the test statistic

and perform the test.6. State the conclusion.

In general, when you make a testable hypothesis, you specify the relationship you expect to find between your IV and the DV.

If you specify the exact direction of the relationship (i.e., longer math tests will increase test anxiety), then you will perform a 1-tailed test.

At other times, you may not know or predict a specific result direction but rather just that performance will change (ie. longer math tests will affect test anxiety), then you will perform a 2-tailed test.

Hypotheses and statistical tests

Hypothesis• NULL Hypothesis: world view or Status quo.• Alternative Hypothesis: Researcher’s theory

– -H0 The average age of a large class is 25.

– -H1 The average age of a large class is different than 25 (two tail)

– - H1 The average age of a large class is less than 25.– Other examples?

Hypothesis Testing about Means# Groups Purpose Test Comment

One Sample and Population comparison

Z-test If is known; and large samples

Sample and Population comparison

t-test If is unknown

Two Comparing two sample means

Z-test If is known; and large samples

Comparing two sample means

t-test If is unknown

Hypothesis testing about means

# Groups Purpose Test Comment

Three or More Comparing multiple sample means

F ANOVA framework

Let’s say you know the value of a particular characteristic in the population (this is uncommon)

- i.e., Computer Industry Satisfaction is normal (mean=100, SD=15) It turns out that we have one CS score for a company X(X = 84)This is a pretty high score. It’s lower than the industry average, but it is “within range.” Based on this one score can I say that X’s score is significantly, different than industry score?

The One-sample Experiment

H1: X has CS lower than industry

H2: X has an equal to or greater CS than industry

Statistical Hypotheses (1-tailed)

Given: i = 100

Null hypothesis (H0): x i

Alternative hypothesis (H1): x < i

H1: X has different CS than industry

H2: X has an equal CS to industry

Statistical Hypotheses (2-tailed)

Given: i = 100

Null hypothesis (H0): x = i

Alternative hypothesis (H1): x ≠ I

1) X indeed has lower CS

2) X has the same CS, but sampling error

produced the smaller mean CS score.

Logic of Statistical Hypothesis TestingWe measured CS on a sample of companies (for the moment N=1) and find that mean CS for the sample of companies is less than the industry mean of 100. Does this mean that X has lower CS?

We could run everyone in the population (not possible), or use inferential statistics to choose between these two alternatives.

Or , the possible explanations are:

Inferential statistics will tell us how likely it would be to get a sample mean like the one we measured given that the null hypothesis (H0) is true.

Logic of Statistical Hypothesis Testing

The logic goes like this:

If it is really UNlikely that we would get a mean score like we did, drawing a sample from the population of the x = 100), then we conclude that our sample did not come from that population, but from a different one. We reject the null hypothesis.

Logic of Statistical Hypothesis Testing

All statistical tests use this logic:- determine the probability that sampling error (random chance) has produced the sample data from a population described by

the null hypothesis (H0).

Let’s look at the z-test to see how this works.

1. We have randomly selected one sample (for the moment N=1)

2. The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale

3. We know the mean of the population of raw scores under some other condition of the independent variable

4. We know the true standard deviation of the population xdescribed by the null hypothesis

Assumptions of the z-test

1. Choose your alpha () level, the criterion you will use to determine whether to accept or reject H0, .05 is typical in psychology/mkt/mgm

2. Identify the region of rejection (1- or 2-tailed)

3. Determine the critical value for your statistic

- for z-test, the critical value is labeled zcrit

Before you compute the test statistic

2-tailed regions

A sampling distribution for H0 showing the region of rejection for = .05 in a 2-tailed z-test.

1-tailed region, above mean

A sampling distribution for H0 showing the region of rejection for = .05 in a 1-tailed z-test.

1-tailed region, below mean

A sampling distribution for H0 showing the region of rejection for = .05 in a 1-tailed z-test where a

decrease in the mean is predicted.

To find zcrit, we will use our z-table to find the z-score that gives us the appropriate proportion of scores in the region of rejection (in the tails of the distribution).

Finding zcrit (2-tailed)

To find zcrit, we will use our z-table to find the z-score that gives us the appropriate proportion of scores in the region of rejection (in the tails of the distribution).

Finding zcrit (1-tailed)

SEM:

You calculate the z-score for your sample mean in a similar way as we did for a single score and it is labeled zobt.

Computing zobt for the test

zobt = 84 - 100 = 16 = -1.0715 15

NSs X

X

X

X

obtt

What does it mean?

You then compare your zobt value to the zcrit value. If zobt is beyond zcrit (more into the tail of the distribution), then we will say that we “reject the null hypothesis”.

Interpreting zobt relative to zcrit

2-tail

1-tail

Note: in this case we fail to reject the null hypothesis. This does not mean the null is true. Indeed, in this case it is almost certainly not true.

Don’t forget this it is really important!!!

We then report that our results were “not significant” or “nonsignificant” and report it like this: z = +1.07, p > .05

Now suppose that we have a sample of 3 companies CS scores.

3 1.73SO,

X X

N

151.73

8.66

and

zobs X X

16

8.66 1.85

You then compare your zobt value to the zcrit value. If zobt is beyond zcrit (more into the tail of the distribution), then we will say that we “reject the null hypothesis”.

Interpreting zobt relative to zcrit

2-tail

1-tail

As a logical consequence, if we have rejected the null hypothesis then we have supported the alternative hypothesis. BUT we have not PROVED anything.

We then report that the difference was “statistically significant” and report it like this:

z = -1.86, p < .05

Type II: accepting the null hypothesis when it is actually false, the probability of making this error can be computed and it is labeled .

Type I: rejecting the null hypothesis when it is actually true, the probability of making this error is equal to , our chosen criterion.

ErrorsSince our statistical procedures are based on probabilities, there is a possibility that our data turned out as they did from chance alone.

We rarely know the population characteristics, particularly the SD, and thus we rarely use the z-test.

If we know the population mean, but not the SD, we must use the t-test rather than the z-test. The logic is identical.

The One-sample t-test

1. We have randomly selected one sample

2. The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale

3. We know the mean of the population of raw scores under some other condition of the independent variable

4. We do not know the true standard deviation of the population described by the null hypothesis so we will estimate it using our sample data.

Assumptions of the t-test

1. Choose your alpha () level, .05 is common

2. Identify the region of rejection (1- or 2-tailed)

3. Determine the critical value for your statistic

- for t-test, the critical value is labeled tcrit

Before you compute the test statistic

You calculate tobt for your sample mean in a similar way as we did for zobt.

Computing tobt for the test

sX sX

N

151.73

8.66

tobs X X

16

8.66 1.85

As with the z-test, for the t-test you will compare tobt to tcrit. So how do you find tcrit? You look in a table of the t-distribution.

Compare tobt relative to tcrit

The t-distribution contains all possible values of t computed for random sample means selected from the population described by the null hypothesis (similar to the z-distribution).

One BIG difference, is that there are many t-distributions for any population, one for every sample size. As N increases, the t-distribution better approximates a normal distribution, until N ~ 120.

Since there are different t-distributions for different sample sizes, we must find tcrit from the appropriate t-distribution.

Finding tcrit

The appropriate t-distribution will be the one that matches our sample size, which we now call degrees of freedom (df).

For the t-test, the degrees of freedom equals N-1, When N-1 is more than 120, the t-distribution is indistinguishable from a true normal distribution and thus use df=.

To find tcrit, use the t-table to find the t for the df that corresponds to your sample size (N-1) for the criterion () you have chosen.

Finding tcrit (2-tailed)

To find tcrit, use the t-table to find the t for the df that corresponds to your sample size (N-1) for the criterion () you have chosen.

Finding tcrit (1-tailed)

If tobt is less than tcrit (away from the area of rejection), then we will say that we “failed to reject the null hypothesis”.

Interpreting tobt relative to tcrit

We then report that our results were “not significant” or “nonsignificant” and report it like this: t (2) = 1.86, p > .05

Modification of calculation when we compare means of two different samples

What is the null? The alternative?

1. Quantitative data.2. One or more categories.3. Independent observations.4. Adequate sample size (at least 10).5. Simple random sample.6. Data in frequency form.7. All observations must be used

Think of types of research questions that can be answeredusing this test

Chi-Square Test Requirements

How do you know the expected frequencies?

1.You hypothesize that they are equal in each group (category)2.Prior knowledge, census data

Category O E (O - E) (O - E)2(O - E)2 /E

yellow 35 30 5 25 0.83red 50 45 5 25 0.56green 30 15 15 225 15blue 10 15 -5 25 1.67white 25 45 -20 400 8.89

Color preference in a car dealership (univariate)

Chi-square=26.95With df=5-1=4