1. Chapters 20, 21
Hypothesis Testing-- Determining if a Result is Different from Expected

2. Example
An Internet hosting company wants to make sure that its connection is running at full speed for a large business at least 90% of the time or more. A sample of 125 tests is collected at randomly chosen times, and the Internet service is running at full speed 82% of the time. Is this enough evidence to suggest that the actual population proportion is not 90%?
This is the type of question we are trying to answer with hypothesis testing.

3. Hypotheses of a Hypothesis Test
Null Hypothesis: The null hypothesis of a hypothesis test is the hypothesis to be tested. We refer to this as H0.
Ex) The population proportion of full speed Internet is 90%.
H0.: p = .90
The null hypothesis is going to be the hypothesis that the actual population proportion equals a certain value. This will be the case even if we are trying to show that this is not the case.

4. The Alternative Hypothesis: a hypothesis considered alternate to the null hypothesis. We use the symbol Ha.
Ex) The population proportion is not 0.90.
Ha: p ≠ 0.90
There are actually three types of Alternative Hypotheses. When we hypothesize that the proportion does not equal a certain value, this is called a two-tailed test.

5. Three Types of Alternative Hypotheses
Two Tailed Test- p ≠ p0
ex) Ha: p ≠ 0.90
One Tailed Tests
1) Left Tailed Test- p < p0
ex) Ha: p < 0.90
2) Right Tailed Test- p > p0
ex) Ha: p > 0.90

6. Choosing The Hypotheses
Null Hypothesis: We always choose the null hypothesis to be a statement that the population proportion equals a certain value.
Alternative Hypothesis: We choose which of the three to use here based on what we are trying to show.
Ex) If we are trying to show that the actual population proportion is less than believed or claimed, we choose the left-tailed alternative hypothesis.

7. Tire Example Ctd.
If we are the company, we may want to test to make sure the proportion does not fall below 90% of the time.
H0: p = .90
Ha: p < .90

8. Performing a Hypothesis Test
How do we perform a hypothesis test?
In other words, how do we determine if the population proportion is actually a different value than expected (or advertised)?
When we look at these situations, we are trying to determine if there is enough evidence to suggest that the actual population proportion is different from the hypothesized proportion.

9. Hypothesis Testing Strategy
To begin, we find what is called a test statistic.
This is simply the z-score:
Sample proportion
Hypothesized pop. proportion
Standard deviation of the sampling dist.

10. Hypothesis Testing Strategy ctd.
We then determine where the test-statistic falls.
Do Not Reject H0
Reject H0
Reject H0
Nonrejection Region
Critical values

11. Critical Values
The critical values are determined by the significance level we want to test at.
Example: If we want to be 95% confident (more to come on this later) in our result, we test at the significance level. This would make our critical values lie at and 1.96.

12. One-tailed Tests Left Tailed Test Right Tailed Test
Nonrejection region
Nonrejection region
Critical value
Rejection Region
Rejection Region
Critical value

13. Internet Example Completed
Hypotheses: H0: p = .90
Ha: p < .90
Significance Level: Assume α = .05
Check Assumptions and Conditions:
Independence
Randomization
10% condition
Success/Failure Condition

14. Tire Example Ctd. Test Statistic:
Critical Value of a left-tailed test with 5% significance: z =

15. Tire Example ctd.
Since the test statistic falls in the rejection region, we should reject the null hypothesis.

16. Two Types of Errors
There are two types of errors that can be made in hypothesis testing.
Type I error: Rejecting a true null hypothesis.
Type II error: Not rejecting a false null hypothesis.
Question: What is the probability of a type I error?

17. Probabilities of Errors
The probability of a type I error is the significance level, , of a hypothesis test.
The smaller the significance level, , the greater the probability of making a type II error, , becomes.

18. The p-value approach to hypothesis testing
The p-value indicates how likely it would be to observe the value obtained for the test statistic if the null hypothesis were true (in other words, if the hypothesized mean was the actual population mean.)
Note: This is a conditional probability.
What does this mean?

19. Interpretation of the p-value
If the p-value is small, then it would be extremely unlikely to obtain that sample mean if the null hypothesis was true.
Thus, for a p-value smaller than the desired significance level, we reject the null hypothesis.
In general, small p-values provide evidence against the null hypothesis.

20. Review of the two-methods
Critical Value Approach
Step 1: State the null and alternative hypotheses.
Step 2: Determine the significance level.
Step 3: Determine the critical value(s).
Step 4: Determine the value of the test statistic.
Step 5: If the test statistic falls in the rejection region, we reject the null hypothesis. Otherwise, we do not.
Step 6: State the conclusion in words.

21. P-value approach Step 1: State the null and alternative hypotheses
Step 2: Determine the significance level.
Step 3: Calculate the value of the test-statistic.
Step 4: Using Table II, calculate the p-value for the test statistic.
Step 5: If reject the null hypothesis. Otherwise, we do not.
Step 6: State your conclusion in words.

22. Example
In a recent survey, 61% of respondents prefer a chocolate variety of ice cream, according to a survey of 1000 randomly selected adults.
Find a 95% confidence interval for the proportion who prefer a chocolate variety.

23. Example ctd.
Test the null hypothesis that half of the people prefer a chocolate variety of ice cream.
One-tailed or two-tailed?

24. Example ctd.
What is the P-value?