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Hypothesis Testing

- Is It Significant?

Questions

- What is a statistical hypothesis?
- What is the null hypothesis? Why is it important

for statistical tests? - Describe the steps in a test of the null

hypothesis. - What are the four kinds of outcome of a

statistical test (compare the sample result to

the state in the population)?

More questions

- What is statistical power?
- What are the factors that influence the power of

a test? - Give a concrete example of a study (describe the

IV and DV) and state one thing you could do to

increase its power.

Decision Making Under Uncertainty

- You have to make decisions even when you are

unsure. School, marriage, therapy, jobs,

whatever. - Statistics provides an approach to decision

making under uncertainty. Sort of decision

making by choosing the same way you would bet.

Maximize expected utility (subjective value). - Comes from agronomy, where they were trying to

decide what strain to plant.

Statistics as a Decision Aid

- Because of uncertainty (have to estimate things),

we will be wrong sometimes. - The point is to be thoughtful about it how many

errors of what kinds? What are the consequences? - Statistics allows us to calculate probabilities

and to base our decisions on those. We choose

(at least partially) the amount and kind of

error. - Hypothesis testing done mostly by convention, but

there is a logic to it.

Statistical Hypotheses

- Statements about characteristics of populations,

denoted H - H normal distribution,
- H N(28,13)
- The hypothesis actually tested is called the

null hypothesis, H0 - E.g.,
- The other hypothesis, assumed true if the null is

false, is the alternative hypothesis, H1 - E.g.,

Testing Statistical Hypotheses - steps

- State the null and alternative hypotheses
- Assume that required to specify the (e.g., SD,

normal distribution, etc.) sampling distribution

of the statistic - Find rejection region of sampling distribution

that place which is not likely if null is true - Collect sample data. Find whether statistic

falls inside or outside the rejection region. If

statistic falls in the rejection region, result

is said to be statistically significant.

Testing Statistical Hypotheses example

- Suppose
- Assume and population is normal, so

sampling distribution of means is known (to be

normal). - Rejection region
- Region (N25)
- We get data
- Conclusion reject null.

Same Example

- Rejection region in z (unit normal)
- Sample result (79) just over the line
- Z (79-75)/2
- Z 2
- 2 gt 1.96

Review

- What is a statistical hypothesis?
- What is the null hypothesis? Why is it important

for statistical tests? - Describe the steps in a test of the null

hypothesis.

Decisions, Decisions

- Based on the data we have, we will make a

decision, e.g., whether means are different. In

the population, the means are really different or

really the same. We will decide if they are the

same or different. We will be either correct or

mistaken.

In the Population

Fire

Sample decision Same Different

Same Right. Null is right, nuts. Type II error. p(Type II)?

Different Type I error. p(Type I) ? Right! Power1-?

Fire Alarm No Yes

Silent Working Yikes!

Goes off False Alarm Working

Conventional Rules

- Set alpha to .05 or .01 (some small value).

Alpha sets Type I error rate. - Choose rejection region that has a probability of

alpha if null is true but some bigger probability

if alternative is true. - Call the result significant beyond the alpha

level (e.g., p lt .05) if the statistic falls in

the rejection region.

Power (1)

- Alpha ( ) sets Type I error rate. We say

different, but really same. - Also have Type II errors. We say same, but really

different. Power is 1- or 1-p(Type II). - It is desirable to have both a small alpha (few

Type I errors) and good power (few Type II

errors), but usually is a trade-off. - Need a specific H1 to figure power.

Power (2)

- Suppose
- Set alpha at .05 and figure region.
- Rejection region is set for alpha .05.

Power (3)

If the bound (141.3) was at the mean of the

second distribution (142), it would cut off 50

percent and Beta and Power would be .50. In this

case, the bound is a bit below the mean. It is

z(141.3-142)/2 -.35 standard errors down. The

area to the right is .36. This means that Beta

is .36 and power is .64.

- 4 Things affect power
- H1, the alternative hypothesis.
- The value and placement of rejection region.
- Sample size.
- Population variance.

Power (4)

The larger the difference in means, the greater

the power. This illustrates the choice of H1.

Power (5)

1 vs. 2 tails rejection region

Rejection Regions

- 1-tailed vs. 2-tailed tests.
- The alternative hypothesis tells the tale

(determines the tails). - If

Nondirectional 2-tails

Directional 1 tail (need to adjust null for

these to be LE or GE).

In practice, most tests are two-tailed. When

you see a 1-tailed test, its usually because it

wouldnt be significant otherwise.

Rejection Regions (2)

- 1-tailed tests have better power on the

hypothesized size. - 1-tailed tests have worse power on the

non-hypothesized side. - When in doubt, use the 2-tailed test.

Power (6)

Sample size and population variability both

affect the size of the standard error of the

mean. Sample size is controlled directly. The

standard deviation is influenced by experimental

control and reliability of measurement.

Review

- What are the four kinds of outcome of a

statistical test (compare the sample result to

the state in the population)? - What is statistical power?
- What are the factors that influence the power of

a test? - Give a concrete example of a study (describe the

IV and DV) and state one thing you could do to

increase its power.

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