1. Section 8-1
Review and Preview

2. Quick Review & Preview
In Chapters 2 and 3 we used “descriptive statistics” when we summarized data using tools such as graphs, and statistics such as the mean and standard deviation.
Methods of inferential statistics use sample data to make an inference or conclusion about a population.
In Chapter 7 we presented methods for estimating a population parameter with a confidence interval.
In this chapter we present the method of hypothesis testing.
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

3. Definitions
In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population.
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

4. Chapter Learning Target:
The main objective of this chapter is to develop the ability to conduct hypothesis tests for claims made about a population proportion p or a population mean .
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

5. Examples of Hypotheses that can be Tested
Genetics: The Genetics & IVF Institute claims that its XSORT method allows couples to increase the probability of having a baby girl. Business: A newspaper headline makes the claim that most workers get their jobs through networking. Medicine: Medical researchers claim that when people with colds are treated with echinacea, the treatment has no effect.
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

6. Examples of Hypotheses that can be Tested
Aircraft Safety: The Federal Aviation Administration claims that the mean weight of an airline passenger (including carry-on baggage) is greater than 185 lb, which it was 20 years ago. Quality Control: When new equipment is used to manufacture aircraft altimeters, the new altimeters are better because the variation in the errors is reduced so that the readings are more consistent. (In many industries, the quality of goods and services can often be improved by reducing variation.)
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

7. Examples of Hypotheses that can be Tested – One you might recognize…
NFL: When a call is made on the field (such as a touchdown), but a “challenge” is made by the other coach to dispute the call made by the referee. Once a call is challenged, the play call is reviewed. The hypothesis would vary depending on the play in question. -- Similar circumstances now apply to other sports such as baseball, hockey, and tennis
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

8. Examples of Hypotheses that can be Tested – A few more…
A hypothesis is put forward that children who take vitamin C are less likely to become ill during flu season than those who do not. A hypothesis test is conducted where a sample group of children is given vitamin C for three months while another group is not.
Most redheads, a hypothesis suggests, are insecure about their hair color. A hypothesis test is conducted wherein redheads are given polygraph testing to determine how they feel about their hair color.
Hypothesis testing of the conception that people with obese parents are likely to become obese themselves suggests that obesity has little to do with genetics.
page 386 of Elementary Statistics, 10th Edition
Various examples are provided below definition box

9. Basics of Hypothesis Testing
Section 8-2
Basics of Hypothesis Testing

11. Null Hypothesis: H0
The null hypothesis (denoted by H0) is a statement that the value of a population parameter (such as proportion or mean) is equal to some claimed value.
We test the null hypothesis directly and either “reject H0” or “fail to reject H0”
Emphasize “equal to”.

12. Alternative Hypothesis: H1
The alternative hypothesis (denoted by H1 or Ha or HA) is the statement that the parameter has a value that somehow differs from the null hypothesis.
The symbolic form of the alternative hypothesis must use one of these symbols: , <, >.
If the problem says phrases like:
At least, No less than,
Greater than or equal to
Use: <
If the problem says phrases like:
At most, No more than,
Less than or equal to
Use: >
Give examples of different ways to word “not equal to,” < and >, such as ‘is different from’, ‘fewer than’, ‘more than’, etc.

13. Example 1: A manager of an airline claims that the mean weight of airline passengers (including carry-on baggage) is at most 195 lb (the current value used by the Federal Aviation Administration). Identify the null hypothesis and the alternative hypothesis.

14. Example 2: An entomologist writes an article in a scientific journal which claims that fewer than 7 in
ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light to identify the null hypothesis and the alternative hypothesis.

15. Example 3: A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. Identify the null hypothesis and the alternative hypothesis.

16. Example 4: A researcher claims that the amounts of acetaminophen in a certain brand of cold tablets have a mean different from the 3.3 mg claimed by the manufacturer. Identify the null hypothesis and the alternative hypothesis.

17. Example 5: A researcher claims that 81% of voters favor tax cuts
Example 5: A researcher claims that 81% of voters favor tax cuts. Identify the null hypothesis and the alternative hypothesis.

18. Test Statistic
The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

19. Test Statistic - Formulas
Test statistic for proportion
Test statistic for mean

20. Example 6: Let’s again consider the claim that the XSORT method of gender selection increases the likelihood of having a baby girl. Preliminary results from a test of the XSORT method of gender selection involved 14 couples who gave birth to 13 girls and 1 boy. Use the given claim and the preliminary results to calculate the value of the test statistic. Use the following hypotheses:
H0: p = 0.5 and H1: p > 0.5.

21. Example 7: The claim is that the proportion of drowning deaths of children attributable to beaches is more than 0.25, and the sample statistics include n = 681 drowning deaths of children with 30% of them attributable to beaches. Use the given claim and the preliminary results to calculate the value of the test statistic.

22. Example 8: The claim is that the proportion of peas with yellow pods is equal to The sample statistics from one of Mendel’s experiments include 580 peas with 152 of them having yellow pods. Use the given claim and the preliminary results to calculate the value of the test statistic.

23. Critical Region
The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis.
Rejection region

24. Significance Level
The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. This is the same  introduced in Section Common choices for  are 0.05, 0.01, and 0.10.

25. Critical Value
A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis. The critical values depend on the nature of the null hypothesis, the sampling distribution that applies, and the significance level . See the previous figure where the critical value of z* = corresponds to a significance level of  = 0.05.
Rejection region

26. Two-tailed Test
When H0 uses “=” and H1 “” (means less than or greater than)
α is divided equally between the two tails of the critical region
Invnorm(______________)
–z* z*
z* = ±critical value (we find these values the same way we found z* in chapter 7)

27. Left-tailed Test
When H0 uses “=” and H1 “<” (If you turn the “less than” symbol into an arrow, it points LEFT)
α is the left tail
Invnorm(______________)
z* = –critical value (we find this value the same way we found z* in chapter 7)

28. Right-tailed Test
When H0 uses “=” and H1 “>” (If you turn the “greater than” symbol into an arrow, it points RIGHT)
α is the right tail
Invnorm(______________)
z* = +critical value (we find this value the same way we found z* in chapter 7)

29. Example 9: Assume that the normal distribution applies and find the critical z* value.
a) α = 0.03 for a two-tailed test.
b) α = 0.03 for a right-tailed test.
c) α = 0.03 for a left-tailed test.

30. Example 10: Assume that the normal distribution applies and find the critical z* value.
a) α = 0.07; H1 is p ≠ 0.12.
b) α = 0.07; H1 is p < 0.12.
c) α = 0.07; H1 is p > 0.12.

31. P-Value
The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.

32. P-Value On Calc: normalcdf (–999999, -z, 0, 1) -z
Critical region in the left tail: P-value = area to the left of the test statistic
On Calc: normalcdf (–999999, -z, 0, 1) -z

33. P-Value z On Calc: normalcdf(z, 999999, 0, 1)
Critical region in the right tail: P-value = area to the right of the test statistic
On Calc: normalcdf(z, , 0, 1) z

34. P-Value - z On Calc: normalcdf (–999999, -z, 0, 1)*2
Critical region in the two tails: P-value = twice area in the tail beyond the test statistic.
- z
On Calc: normalcdf (–999999, -z, 0, 1)*2

35. P-Value
The null hypothesis is rejected if the P-value is very small, such as 0.05 or less.
Here is a memory tool useful for interpreting the P-value:
If the P is low, the null must go.
If the P is high, the null will fly.

36. Or stated another way…

37. Or one other way…
Reject that Ho

38. Procedure for Finding P-Values

39. Don’t confuse a P-value with a proportion p. Know this distinction:
Caution
Don’t confuse a P-value with a proportion p.
Know this distinction:
P-value = probability of getting a test statistic at least as extreme as the one representing sample data
p = population proportion
page 394 of Elementary Statistics, 10th Edition

40. Example 11: Consider the claim that with the XSORT method of gender selection, the likelihood of having a baby girl is different from p = 0.5, and use the test statistic z = 3.21 found from 13 girls in 14 births.
a) Determine whether the given conditions result in a critical region in the right tail, left tail, or two tails.
b) Find the P-value.
page 394 of Elementary Statistics, 10th Edition

41. Example 12: Use the given information to find the P-value
Example 12: Use the given information to find the P-value. The test statistic in a right-tailed test is z = 1.57.
page 394 of Elementary Statistics, 10th Edition

42. Example 13: Use the given information to find the P-value
Example 13: Use the given information to find the P-value. The test statistic in a left-tailed test is z = –1.84.
page 394 of Elementary Statistics, 10th Edition

43. Example 14: Use the given information to find the P-value
Example 14: Use the given information to find the P-value. The test statistic in a two-tailed test is z = –1.76.
page 394 of Elementary Statistics, 10th Edition

44. H1: p ≠ 0.8, the test statistic is z = 2.01.
Example 15: Use the given information to find the P-value. With
H1: p ≠ 0.8, the test statistic is z = 2.01.
page 394 of Elementary Statistics, 10th Edition

45. H1: p < 0.8, the test statistic is z = –1.91.
Example 16: Use the given information to find the P-value. With
H1: p < 0.8, the test statistic is z = –1.91.
page 394 of Elementary Statistics, 10th Edition

46. H1: p > 0.8, the test statistic is z = 2.49.
Example 17: Use the given information to find the P-value. With
H1: p > 0.8, the test statistic is z = 2.49.
page 394 of Elementary Statistics, 10th Edition

47. Types of Hypothesis Tests: Two-tailed, Left-tailed, Right-tailed
The tails in a distribution are the extreme regions bounded by critical values.
Determinations of P-values and critical values are affected by whether a critical region is in two tails, the left tail, or the right tail. It therefore becomes important to correctly characterize a hypothesis test as two-tailed, left-tailed, or right-tailed.
page 394 of Elementary Statistics, 10th Edition

48. Two-tailed Test
H0: =
H1: 
α is divided equally between the two tails of the critical region
Means less than or greater than

49. Left-tailed Test
H0: =
H1: <
𝛼 is in the left tail
Points Left

50. Right-tailed Test
H0: =
H1: >
𝛼 is in the right tail
Points Right

51. Conclusions in Hypothesis Testing
We always test the null hypothesis. The initial conclusion will always be one of the following:
1. Reject the null hypothesis.
2. Fail to reject the null hypothesis.
page 394 of Elementary Statistics, 10th Edition

52. Decision Criterion P-value method: Using the significance level :
If P-value   , reject H0.
If P-value >  , fail to reject H0.

53. Decision Criterion Traditional method:
If the test statistic falls within the critical region, reject H0.
If the test statistic does not fall within the critical region, fail to reject H0.

54. Wording of Final Conclusion
There is enough evidence to suggest that the … (alternative hypothesis in context)
There is not enough evidence to suggest that the … (alternative hypothesis in context)

55. Initial conclusion: Fail to reject the null hypothesis.
Example 18: State the final conclusion in simple nontechnical terms.
Alternative hypothesis: The percentage of blue M&Ms is greater than 5%.
Initial conclusion: Fail to reject the null hypothesis.
page 394 of Elementary Statistics, 10th Edition

56. Example 19: An entomologist writes an article in a scientific journal which claims that fewer than 12 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.
page 394 of Elementary Statistics, 10th Edition

57. Initial conclusion: Reject the null hypothesis.
Example 20: State the final conclusion in simple nontechnical terms.
Alternative hypothesis: The percentage of Americans who know their credit score is not equal to 20%.
Initial conclusion: Reject the null hypothesis.
page 394 of Elementary Statistics, 10th Edition

58. One Final Meme 