Comprehensive Guide to Hypothesis Testing: Defining, Testing, and Decision Making Process

1. Chapter Seven Hypothesis Testing with ONE Sample

2. Section 7.1 Introduction to Hypothesis Testing

3. Hypothesis Tests • … A process that uses sample statistics to test a claim about a population parameter. • Test includes: • Stating a NULL and an ALTERNATIVE Hypothesis. • Determining whether to REJECT or to NOT REJECT the Null Hypothesis. (If the Null is rejected, that means the Alternative must be true.)

4. Stating a Hypothesis • The Null Hypothesis (H0) is a statistical hypothesis that contains some statement of equality, such as =, <, or > • The Alternative Hypothesis (Ha) is the complement of the null hypothesis. It contains a statement of inequality, such as ≠, <, or >

5. Left, Right, or Two-Tailed Tests • If the Alternative Hypotheses, Ha , includes <, it is considered a LEFT TAILED test. • If the Alternative Hypotheses, Ha , includes >, it is considered a RIGHT TAILED test. • If the Alternative Hypotheses, Ha , includes ≠, it is considered a TWO TAILED test.

6. EX: State the Null and Alternative Hypotheses. 26. As stated by a company’s shipping department, the number of shipping errors per mission shipments has a standard deviation that is less than 3. 28. A state park claims that the mean height of oak trees in the park is at least 85 feet.

7. Types of Errors When doing a test, you will decide whether to reject or not reject the null hypothesis. Since the decision is based on SAMPLE data, there is a possibility the decision will be wrong. Type I error: the null hypothesis is rejected when it is true. Type II error: the null hypothesis is not rejected when it is false.

8. 4 possible outcomes…

9. Level of Significance • The level of significance is the maximum allowed probability of making a Type I error. It is denoted by the lowercase Greek letter alpha. • The probability of making a Type II error is denoted by the lowercase Greek letter beta.

10. p-Values • If the null hypothesis is true, a p-Value of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. • The p-Value is connected to the area under the curve to the left and/or right on the normal curve.

11. Making and Interpreting your Decision • Decision Rule based on the p-Value Compare the p-Value with alpha. • If p < alpha, reject H0 • If p > alpha, do not reject H0

12. General Steps for Hypothesis Testing • State the null and alternative hypotheses. • Specify the level of significance. • Sketch the curve. • Find the standardized statistic add to sketch and shade. (usually z or t-score) • Find the p-Value • Compare p-Value to alpha to make the decision. • Write a statement to interpret the decision in context of the original claim.

13. Section 7.2 Hypothesis Testing for the MEAN (Large Samples)

14. Using p-Value to Make Decisions • Decision Rule based on the p-Value Compare the p-Value with alpha. • If p < alpha, reject H0 • If p > alpha, do not reject H0

15. Finding the p-Value for a Hypothesis Test – using the table • To find p-Value • Left tailed: p = area in the left tail • Right tailed: p = area in the right tail • Two Tailed: p = 2(area in one of the tails) This section we’ll be finding the z-values and using the standard normal table.

16. Find the p-value. Decide whether to reject or not reject the null hypothesis • 4. Left tailed test, z = -1.55, alpha = 0.05 • 8. Two tailed test, z = 1.23, alpha = 0.10

17. Using p-Values for a z-Test • Z-Test used when the population is normal, δ is known, and n is at least 30. If n is more than 30, we can use s for δ.

18. Guidelines – using the p-value • 1. find H0 and Ha • 2. identify alpha • 3. find z • 4. find area that corresponds to z (the p-value) • 5. compare p-value to alpha • 6. make decision • 7. interpret decision

19. 30. A manufacturer of sprinkler systems designed for fire protection claims the average activating temperature is at least 135oF. To test this claim, you randomly select a sample of 32 systems and find mean = 133, and s = 3.3. At alpha = 0.10, do you have enough evidence to reject the manufacturer’s claim?

20. Rejection Regions & Critical Values • The Critical value (z0) is the z-score that corresponds to the level of significance (alpha) • Z0 separates the rejection region from the non-rejection region • Sketch a normal curve and shade the rejection region. (Left, right, or two tailed)

21. Find z0 and shade rejection region • 18. Right tailed test, alpha = 0.08 • 22. Two tailed test, alpha = 0.10

22. Guidelines – using rejection regions • 1. find H0 and Ha • 2. identify alpha • 3. find z0 – the critical value(s) • 4. shade the rejection region(s) • 5. find z • 6. make decision (Is z in the rejection region?) • 7. interpret decision

23. 38. A fast food restaurant estimates that the mean sodium content in one of its breakfast sandwiches is no more than 920 milligrams. A random sample of 44 sandwiches has a mean sodium content of 925 with s = 18. At alpha = 0.10, do you have enough evidence to reject the restaurant’s claim?