HYPOTHESIS TESTING: A FORM OF STATISTICAL INFERENCE

Mrs. WatkinsAP Statistics

Chapters 23,20,21

What is a hypothesis test?

• Hypothesis Testing: Method for using sample data to decide between 2 competing claims about a population parameter (mean or proportion)

What question do such tests answer?

Is our finding due to chance or is it likely that something about the population seems to have changed?

Why Statistical Inference?

The only way to “prove” anything is to use entire population, which is not possible.

So, we use INFERENCE to make decisions about a population, based on a sample

EXAMPLE:

• A new cold medicine claims to reduce the amount of time a person suffers with a cold. A random sample of 25 people took the new medicine when they felt the onset of a cold and continue to take it twice a day until theyfelt better. The average time these people took the medication was 5.2 days with a standarddeviation of 1.4 days. The typical time a person suffers with a cold is said to be one week.

Questions from our cold study:

What is the difference between the population mean and the sample mean? 1.8 days

Is this difference likely to be due to chance?

How could we compute how likely it is to see a mean of 5.2 when we are expecting a mean of 7 days? Use a z score!

z = 7 – 5.2_ = 6.43 1.4/√25

probability of this is nearly 0…so unlikely

Hypotheses: Ho: μ = 7 (the status quo of cold duration)Ha: μ < 7 (what we hope to be true about the

new medication)

Our evidence suggests that Ha is more likely to be true.

Writing Hypotheses:

Statistical Hypothesis: a claim or statement about the value of the population parameter

2 Hypotheses:• Null Hypothesis: claim that is assumed to be

true—usually based on past research Noted Ho

• Alternative Hypothesis: competing claim based on a new sample suggesting that a change has occurred Noted Ha

Kinds of Tests:

Two tailed: Ho: μ = 7Ha: μ ≠ 7

Right tailed: Ho: μ = 7 Ha: μ > 7

Left tailed: Ho: μ = 7 Ha: μ < 7

Hypotheses Example 1:

A medical researcher wants to know if a new medicine will have an effect on a patient’s pulse rate. He knows that the mean pulse rate for this population is 82 beats per minute:

Ho: μ = 82

Ha: μ ≠ 82

Hypotheses Example 2:

A chemist invents an additive to increase the life of an automobile battery. The mean lifetimes of a typical car battery is 36 months.

Ho:μ = 36Ha:μ > 36

Hypotheses Example 3:

An educational research group is investigating the effects of poverty on elementary school reading levels. Prior research suggests that only 46% of children from poor families achieve grade level reading by third grade

Ho: p = 0.46Ha: p ≠ 0.46

Hypotheses Example 4:

A cancer research team has been given the task of evaluating a new laser treatment for tumors. The current standard treatment is costly and has a success rate of 0.30.

Ho: p = 0.30Ha: p > 0.30

Statistical Significance:

The results of an experiment or observational study are too “different” from the established population parameter to have occurred simply due to chance….

Something else must be going on…..

ASSIGNMENT:

Now go on-line and watch this video carefully for good example of hypothesis testing in use: http://www.learner.org/courses/againstallodds/unitpages/unit25.html

α = rejection region

α is the rejection region on the normal curve, accepted to be the highest probability that cause you to uphold the Ho.

RESULTS OF HYPOTHESES TESTS

Let’s assume α = 0.05.If p < α, then we reject Ho.

The sample result is too unlikely to have happened due to chance, so the Ho is overturned.

If p > α, then we fail to reject Ho.The sample result could have

happened due to chance, so the Ho is upheld.

What does p value mean?

The p value is the probability (based on z or t curve) of seeing a sample mean of this value or more extreme if the Ho is really true.

If p value is low, then the Ho must not be true. The sample data suggests that the status quo has changed.

Conclusions of Hypothesis Tests

Rejecting Ho = Statistically significant change

Failing to reject Ho=Difference between sample mean and Ho mean was not statistically significant.

Testing about MeansWhen investigating whether a claim about a MEAN is correct, you have to decide whether to do a t test or a z test.

Z test: if you know pop. standard deviationT test: if you know sample standard deviation

HYPOTHESIS TESTS

H: HypothesesA: AssumptionsT: Test and Test StatisticP: P valueI: Interpretation of p valueC: Conclusion

HYPOTHESIS TESTING FOR PROPORTIONS

EXAMPLE

A newspaper article from 5 years ago claimed that 9.5% of college students seriously considered suicide sometime during the previous year. If a sample from this year consisted of 1,000 students and 144 claimed that they had seriously considered suicide, is there evidence to suggest that the proportion has increased?

DRAW THE MODEL OF THE SAMPLING DISTRIBUTION OF THE PROPORTION

Hypotheses

Null Hypothesis: Ho : p = 0.095(the stated claim about the population

proportion)Alternative Hypothesis:

Ha: p > 0.095Ha: p < 0.095Ha: p ≠ 0.095

Z Proportion Test

There are no t tests for proportions, only z.Note that we are using p not for standard deviation of distribution.

Test Statistic: z = P value: use normal cdf

Assumptions:

1. Random Sample or Random Assignment2. Large enough to assume normal model:

n p > 10n q > 10

Note that we are using p not for verifying normality assumption.

EXAMPLE: DO HATPIC

An educator claims the dropout rate in Ohio schools is 15%. Last year, 280 seniors from a random sample of 2000 seniors withdrew from school.

At α = 0.05, can the claim of 15% be supported or is the proportion statistically significantly different?