Hypothesis Testing Overview: Lecture 10

1. Lecture 10 Dan Piett STAT 211-019 West Virginia University

2. Exam 2 Results

3. Overview • Introduction to Hypothesis Testing • Hypothesis Tests on the population mean • Hypothesis Tests on the population proportion

4. Section 11.1 Introduction to Hypothesis Testing

5. Hypothesis Testing • We 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 testing • Confidence intervals • Give predictions on bounds which we believe a proportion will fall • Hypothesis Testing • Deciding which of two hypotheses are more likely

6. 4 (7) Steps to Hypothesis Testing • State the Null and Alternative Hypotheses, and the Significance Level • State the Null Hypothesis • State the Alternative Hypothesis • State the Significance Level (alpha) • Calculate the Test Statistic • Calculate the Test Statistic • Find the p-value • Find the p-value • Make a Decision and State your Conclusion • Make a Decision • State the Conclusion

7. Step 1: Stating The Null Hypothesis • Hypothesis • A statement about a population parameter. • A hypothesis is either true or false • Hypotheses are mutually exclusive • The Null Hypothesis (H0) • Typically expresses the idea of “no difference”, “no change” or equality” • Contains an equal sign • Example: • H0 : µ = 100 • H0 : p = .35

8. 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 Hypothesis • Examples: • H1 : µ < 100 • H1 : µ > 100 • HA : p ≠ .35

9. 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 Hypothesis • H0 : µ = 21 • Possible Alternative Hypotheses • Less than • HA : µ < 21 • More than • HA : µ > 21 • Not equal to • HA : µ ≠ 21

10. Step 3: The Significance Level • The 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

11. 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-value • The 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 decision • Two sided alternative p-values are 2x as large as 1 sided

12. Steps 6/7: Making a Decision • Making our Decision • If our p-value is < our significance level • If the p-value is low, the H0 has got to go • We reject our Null Hypothesis • If the p-value is > our significance level • We fail to reject our Null Hypothesis • We DO NOT accept the Null Hypothesis • Stating our Conclusion • Example: We have enough evidence at the .05 level to conclude that the mean age of all college students is not equal to 21.

13. 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.

14. Section 11.2 Hypothesis Tests on the Population Mean

15. Hypothesis Tests on the Pop. Mean • The first hypothesis test we will look at is testing for a value of the population mean • We 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 intervals • Z if n ≥ 20 or we know the population standard deviation (sigma) • T otherwise

16. Hypothesis Test on µ (Lg. Sample or we know sigma) • H0: µ = # • HA: µ < # or µ > # or µ ≠ # • Alpha is .05 if not specified • Test Statistic = Z = • 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|) • Our decision rule will be to reject H0 if p-value < alpha • 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

17. 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

18. Hypothesis Test on µ (Sm. Sample) • H0: µ = # • HA: µ < # or µ > # or µ ≠ # • Alpha is .05 if not specified • Test Statistic = T = • 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|) • Our decision rule will be to reject H0 if p-value < alpha • We have (do not have) enough evidence at the .05 level to conclude that the mean ______ is (<, >, ≠) # Requires independent random samples

19. Example Automobile 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

20. Section 11.3 Hypothesis Tests on the Population Proportion

21. Hypothesis Tests on p • We can test hypotheses involved in binomial parameter, p. • Similar Rules apply to when we computed confidence intervals on p • Binomial Experiment • np > 5 • n(1-p)>5

22. Hypothesis Tests on p • H0: p = # • HA: p < # or p > # or p ≠ # • Alpha is .05 if not specified • Test Statistic = Z = • 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|) • Our decision rule will be to reject H0 if p-value < alpha • 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

23. Example • The 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.