Dan Piett STAT West Virginia University Lecture 10

Exam 2 Results

Overview Introduction to Hypothesis Testing Hypothesis Tests on the population mean Hypothesis Tests on the population proportion

Section 11.1 Introduction to Hypothesis Testing

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

4 (7) Steps to Hypothesis Testing 1. State the Null and Alternative Hypotheses, and the Significance Level 1. State the Null Hypothesis 2. State the Alternative Hypothesis 3. State the Significance Level (alpha) 2. Calculate the Test Statistic 4. Calculate the Test Statistic 3. Find the p-value 5. Find the p-value 4. Make a Decision and State your Conclusion 6. Make a Decision 7. State the Conclusion

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 (H 0 ) Typically expresses the idea of “no difference”, “no change” or equality” Contains an equal sign Example: H 0 : µ = 100 H 0 : p =.35

Step 2: State the Alternative Hyp. The Alternative Hypothesis (H 1 or H A ) Expresses the idea of a difference, change, or inequality Contains an inequality symbol (, ≠) 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: H 1 : µ < 100 H 1 : µ > 100 H A : 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 Hypothesis H 0 : µ = 21 Possible Alternative Hypotheses Less than H A : µ < 21 More than H A : µ > 21 Not equal to H A : µ ≠ 21

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 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-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

Steps 6/7: Making a Decision Making our Decision If our p-value is < our significance level If the p-value is low, the H 0 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.

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. 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

Hypothesis Test on µ (Lg. Sample or we know sigma) 1. H 0 : µ = # 2. H A : 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 H 0 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. H 0 : µ = # 2. H A : 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 |T|) For ≠ alternative: 2*P(t>|T|) 6. Our decision rule will be to reject H 0 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

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

Section 11.3 Hypothesis Tests on the Population Proportion

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

Hypothesis Tests on p 1. H 0 : p = # 2. H A : 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 H 0 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

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.