Dan PiettSTAT 211-019

West Virginia University

Lecture 10

Exam 2 Results

OverviewIntroduction to Hypothesis TestingHypothesis Tests on the population meanHypothesis Tests on the population

proportion

Section 11.1

Introduction to Hypothesis Testing

Hypothesis TestingWe have looked at one way of making

inferences about population parameters (µ, p, etc)We did this using confidence intervals calculated

with sample statistics (x-bar, p-hat, etc)We will look into a method known as

hypothesis testingConfidence intervals

Give predictions on bounds which we believe a proportion will fall

Hypothesis Testing Deciding which of two hypotheses are more likely

4 (7) Steps to Hypothesis Testing1. State the Null and Alternative Hypotheses, and the

Significance Level1. State the Null Hypothesis2. State the Alternative Hypothesis3. State the Significance Level (alpha)

2. Calculate the Test Statistic4. Calculate the Test Statistic

3. Find the p-value5. Find the p-value

4. Make a Decision and State your Conclusion6. Make a Decision7. State the Conclusion

Step 1: Stating The Null HypothesisHypothesis

A statement about a population parameter. A hypothesis is either true or falseHypotheses are mutually exclusive

The Null Hypothesis (H0)Typically expresses the idea of “no

difference”, “no change” or equality”Contains an equal signExample:

H0 : µ = 100H0 : p = .35

Step 2: State the Alternative Hyp.The Alternative Hypothesis (H1 or HA)

Expresses the idea of a difference, change, or inequality

Contains an inequality symbol (<, >, ≠)< and > are considered one sided alternatives≠ is considered a two sided alternative

We will talk more about 2 sided alternatives in another class

Also known as the Research HypothesisExamples:

H1 : µ < 100H1 : µ > 100HA : p ≠ .35

Larger Example A researcher is interested in the mean age of all college

students. He believes that the mean age of all college students is ______ 21.

Null HypothesisH0 : µ = 21

Possible Alternative HypothesesLess than

HA : µ < 21More than

HA : µ > 21Not equal to

HA : µ ≠ 21

Step 3: The Significance LevelThe Significance Level ( )

This value determines how far our data must be from our Null Hypothesis to be considered Significantly Different

This relates to our confidence levels from confidence intervals

Common values for alpha.1.05.01.001

We will see how these work in a later step

Step 4/5: Calculating the Test Statistic and finding the p-value Test Statistic

In this step we will find a value (Z or t) that will be used to determine our p value.

The formula for Z or t is determined by which type of hypothesis test we are interested in. We will cover them individually in this lecture and subsequent lectures

P-valueThe p-value is defined as the probability of getting a

value as extreme or more extreme given that our null hypothesis is true

Our p-value will be calculated with a table using our Test Statistic.

We will use our p-value to make our decisionTwo sided alternative p-values are 2x as large as 1 sided

Steps 6/7: Making a DecisionMaking our Decision

If our p-value is < our significance levelIf the p-value is low, the H0 has got to goWe reject our Null Hypothesis

If the p-value is > our significance levelWe fail to reject our Null HypothesisWe DO NOT accept the Null Hypothesis

Stating our ConclusionExample: We have enough evidence at

the .05 level to conclude that the mean age of all college students is not equal to 21.

Summing it up:Steps 1, 2, 3, 6, 7 are very generic the

same for nearly all hypothesis test procedures that will be covered in this class.

The major differences will occur in Step 4, and slight differences will occur in our hypotheses and tables used in our calculation of the p-value.

We will now look into some different types of hypothesis tests.

Section 11.2

Hypothesis Tests on the Population Mean

Hypothesis Tests on the Pop. MeanThe first hypothesis test we will look at is

testing for a value of the population meanWe are interested if the population mean is

different (or >, or <) then some predetermined value.

Our test statistic will be of the formula:The same rules apply as for confidence

intervalsZ if n ≥ 20 or we know the population

standard deviation (sigma)T otherwise

Hypothesis Test on µ (Lg. Sample or we know sigma)1. H0: µ = #

2. HA: µ < # or µ > # or µ ≠ #

3. Alpha is .05 if not specified

4. Test Statistic = Z =

5. P-value will come from the normal dist. Table For > alternative: P(z>Z) For < alternative: P(z<Z) For ≠ alternative:2*P(z>|Z|)

6. Our decision rule will be to reject H0 if p-value < alpha

7. We have (do not have) enough evidence at the .05 level to conclude that the mean ______ is (<, >, ≠) #

Requires a large sample size or the population stdv. Also requires independent random samples

Example:A company owns a fleet of cars with mean

MPG known to be 30. This company will use a gasoline additive only if the additive increases gasoline MPG. A sample of 36 cars used the additive. The sample mean was calculated to be 31.3 miles with a standard deviation of 7 miles.

Does the additive significantly increase MPG. Use alpha = .10

Hypothesis Test on µ (Sm. Sample)1. H0: µ = #

2. HA: µ < # or µ > # or µ ≠ #

3. Alpha is .05 if not specified

4. Test Statistic = T =

5. P-value will come from the t-dist. Table with df = n-1 For > alternative: P(t>|T|) For < alternative: P(t>|T|) For ≠ alternative: 2*P(t>|T|)

6. Our decision rule will be to reject H0 if p-value < alpha

7. We have (do not have) enough evidence at the .05 level to conclude that the mean ______ is (<, >, ≠) #

Requires independent random samples

ExampleAutomobile exhaust contains an average of

90 parts per million of carbon monoxide. A new pollution control device is placed on ten randomly selected cars. For these 10 cars, mean carbon monoxide emission was 75 ppm with a standard deviation of 20 ppm.

Does the new pollution control device significantly reduce carbon monoxide emission. Use alpha = .05

Section 11.3

Hypothesis Tests on the Population Proportion

Hypothesis Tests on pWe can test hypotheses involved in

binomial parameter, p.Similar Rules apply to when we computed

confidence intervals on pBinomial Experimentnp > 5n(1-p)>5

Hypothesis Tests on p1. H0: p = #

2. HA: p < # or p > # or p ≠ #

3. Alpha is .05 if not specified

4. Test Statistic = Z =

5. P-value will come from the normal dist. Table For > alternative: P(z>Z) For < alternative: P(z<Z) For ≠ alternative:2*P(z>|Z|)

6. Our decision rule will be to reject H0 if p-value < alpha

7. We have (do not have) enough evidence at the .05 level to conclude that the proportion ______ is (<, >, ≠) #

Requires np>5, n(1-p)>5. Also requires independent random samples

ExampleThe West Virginia Dept. of Revenue stated

that 20% of West Virginians have income below the “poverty level.” I suspect that this percentage is lower than 20% in Mon. County. I obtained a random sample of 400 residents and find that 70 are living below the “poverty line”. Does the data support my suspicion that less than 20% of Mon. County residents live below the poverty level. Use a significance level of .05.