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Chapter 7

Chapter 7. Hypothesis Testing with One Sample. Chapter Outline. 7.1 Introduction to Hypothesis Testing 7.2 Hypothesis Testing for the Mean (Large Samples) 7.3 Hypothesis Testing for the Mean (Small Samples) 7.4 Hypothesis Testing for Proportions

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Chapter 7

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  1. Chapter 7 Hypothesis Testing with One Sample

  2. Chapter Outline • 7.1 Introduction to Hypothesis Testing • 7.2 Hypothesis Testing for the Mean (Large Samples) • 7.3 Hypothesis Testing for the Mean (Small Samples) • 7.4 Hypothesis Testing for Proportions • 7.5 Hypothesis Testing for Variance and Standard Deviation

  3. Section 7.1 Introduction to Hypothesis Testing

  4. Section 7.1 Objectives • State a null hypothesis and an alternative hypothesis • Identify type I and type I errors and interpret the level of significance • Determine whether to use a one-tailed or two-tailed statistical test and find a p-value • Make and interpret a decision based on the results of a statistical test • Write a claim for a hypothesis test

  5. Hypothesis Tests Hypothesis test • A process that uses sample statistics to test a claim about the value of a population parameter. • For example: An automobile manufacturer advertises that its new hybrid car has a mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the sample mean differs enough from the advertised mean, you can decide the advertisement is wrong.

  6. Hypothesis Tests Statisticalhypothesis • A statement, or claim, about a population parameter. • Need a pair of hypotheses • one that represents the claim • the other, its complement • When one of these hypotheses is false, the other must be true.

  7. Stating a Hypothesis Null hypothesis • A statistical hypothesis that contains a statement of equality such as , =, or . • Denoted H0 read “H subzero” or “Hnaught.” Alternative hypothesis • A statement of inequality such as >, , or <. • Must be true if H0 is false. • Denoted Ha read “H sub-a.” complementary statements

  8. Stating a Hypothesis • To write the null and alternative hypotheses, translate the claim made about the population parameter from a verbal statement to a mathematical statement. • Then write its complement. H0: μ ≤ k Ha: μ > k H0: μ ≥ k Ha: μ < k H0: μ = k Ha: μ ≠ k • Regardless of which pair of hypotheses you use, you always assume μ = k and examine the sampling distribution on the basis of this assumption.

  9. Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. A university publicizes that the proportion of its students who graduate in 4 years is 82%. Solution: H0: Ha: p = 0.82 (Claim) Equality condition Complement of H0 p ≠ 0.82

  10. Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute. Solution: H0: Ha: Complement of Ha μ ≥ 2.5 gallons per minute Inequality condition (Claim) μ < 2.5 gallons per minute

  11. Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. A cereal company advertises that the mean weight of the contents of its 20-ounce size cereal boxes is more than 20 ounces. Solution: H0: Ha: μ ≤ 20 ounces Complement of Ha Inequality condition μ > 20 ounces (Claim)

  12. Types of Errors • No matter which hypothesis represents the claim, always begin the hypothesis test assuming that the equality condition in the null hypothesis is true. • At the end of the test, one of two decisions will be made: • reject the null hypothesis • fail to reject the null hypothesis • Because your decision is based on a sample, there is the possibility of making the wrong decision.

  13. Types of Errors • A type I error occurs if the null hypothesis is rejected when it is true. • A type II error occurs if the null hypothesis is not rejected when it is false.

  14. Hypothesis testing is sometimes compared to the legal system used in the United States. Under this system, the following steps are used. 1. A carefully worded accusation is written. 2. The defendant is assumed innocent ( H 0 )  until proven guilty. The burden of proof lies with the prosecution. If the evidence is not strong enough, there is no conviction a “not guilty” verdict does not prove that a defendant is innocent. 3. The evidence needs to be conclusive beyond a reasonable doubt. The system assumes that more harm is done by convicting the innocent (type I error) than by not convicting the guilty (type II error).

  15. Example: Identifying Type I and Type II Errors The USDA limit for salmonella contamination for chicken is 20%. A meat inspector reports that the chicken produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the meat inspector’s claim is true. When will a type I or type II error occur? Which is more serious? (Source: United States Department of Agriculture)

  16. Solution: Identifying Type I and Type II Errors Let p represent the proportion of chicken that is contaminated. H0: Ha: Hypotheses: p ≤ 0.2 (Claim) p > 0.2 Chicken meets USDA limits. Chicken exceeds USDA limits. p H0: p≤ 0.20 H0: p> 0.20 0.16 0.18 0.20 0.22 0.24

  17. Solution: Identifying Type I and Type II Errors Hypotheses: H0: Ha: p ≤ 0.2 (Claim) p > 0.2 A type I error is rejecting H0 when it is true. The actual proportion of contaminated chicken is less than or equal to 0.2, but you decide to reject H0. A type II error is failing to reject H0 when it is false. The actual proportion of contaminated chicken is greater than 0.2, but you do not reject H0.

  18. Solution: Identifying Type I and Type II Errors Hypotheses: H0: Ha: p ≤ 0.2 (Claim) p > 0.2 • With a type I error, you might create a health scare and hurt the sales of chicken producers who were actually meeting the USDA limits. • With a type II error, you could be allowing chicken that exceeded the USDA contamination limit to be sold to consumers. • A type II error could result in sickness or even death.

  19. Level of Significance Level of significance • Your maximum allowable probability of making a type I error. • Denoted by , the lowercase Greek letter alpha. • By setting the level of significance at a small value, you are saying that you want the probability of rejecting a true null hypothesis to be small. • Commonly used levels of significance: •  = 0.10  = 0.05  = 0.01 • P(type II error) = β (beta)

  20. Population parameter Test statistic Standardized test statistic z (Section 7.2 n 30) μ t (Section 7.3 n < 30) p z (Section 7.4) s2 χ2 (Section 7.5) Statistical Tests • After stating the null and alternative hypotheses and specifying the level of significance, a random sample is taken from the population and sample statistics are calculated. • The statistic that is compared with the parameter in the null hypothesis is called the test statistic. σ2

  21. P-values P-value (or probabilityvalue) • The probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. • Depends on the nature of the test.

  22. Nature of the Test • Three types of hypothesis tests • left-tailed test • right-tailed test • two-tailed test • The type of test depends on the region of the sampling distribution that favors a rejection of H0. • This region is indicated by the alternative hypothesis.

  23. P is the area to the left of the test statistic. z -3 -2 -1 0 1 2 3 Test statistic Left-tailed Test • The alternative hypothesis Ha contains the less-than inequality symbol (<). H0: μk Ha: μ< k

  24. P is the area to the right of the test statistic. 0 1 2 3 -3 -2 -1 z Test statistic Right-tailed Test • The alternative hypothesis Ha contains the greater-than inequality symbol (>). H0: μ≤k Ha: μ> k

  25. P is twice the area to the left of the negative test statistic. P is twice the area to the right of the positive test statistic. 0 1 2 3 -3 -2 -1 z Test statistic Test statistic Two-tailed Test • The alternative hypothesis Ha contains the not equal inequality symbol (≠). Each tail has an area of ½P. H0: μ= k Ha: μ k

  26. Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. • A university publicizes that the proportion of its students who graduate in 4 years is 82%. Solution: p = 0.82 H0: Ha: ½ P-value area ½ P-value area p ≠ 0.82 z Two-tailed test -z 0 z

  27. Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. • A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute. Solution: P-value area μ ≥ 2.5 gpm H0: Ha: μ < 2.5 gpm z -z 0 Left-tailed test

  28. Example: Identifying The Nature of a Test For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value. • A cereal company advertises that the mean weight of the contents of its 20-ounce size cereal boxes is more than 20 ounces. Solution: P-value area μ ≤ 20 oz H0: Ha: μ > 20 oz z z 0 Right-tailed test

  29. Making a Decision Decision Rule Based on P-value • Compare the P-value with . • If P , then reject H0. • If P> , then fail to reject H0.

  30. Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0? • H0(Claim): A university publicizes that the proportion of its students who graduate in 4 years is 82%.

  31. Solution: Interpreting a Decision • The claim is represented by H0. • If you reject H0 you should conclude “there is sufficient evidence to indicate that the university’s claim is false.” • If you fail to reject H0, you should conclude “there is insufficient evidence to indicate that the university’s claim (of a four-year graduation rate of 82%) is false.”

  32. Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0? • Ha(Claim): Consumer Reports states that the mean stopping distance (on a dry surface) for a Honda Civic is less than 136 feet. • Solution: • The claim is represented by Ha. • H0 is “the mean stopping distance…is greater than or equal to 136 feet.”

  33. Solution: Interpreting a Decision • If you reject H0 you should conclude “there is enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet.” • If you fail to reject H0, you should conclude “there is not enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet.”

  34. z 0 z 0 Test statistic Steps for Hypothesis Testing • State the claim mathematically and verbally. Identify the null and alternative hypotheses. • H0: ? Ha: ? • Specify the level of significance. • α = ? • Determine the standardized sampling distribution and draw its graph. • Calculate the test statisticand its standardized value.Add it to your sketch. This sampling distribution is based on the assumption that H0 is true.

  35. No Yes Steps for Hypothesis Testing • Find the P-value. • Use the following decision rule. • Write a statement to interpret the decision in the context of the original claim. Is the P-value less than or equal to the level of significance? Fail to reject H0. Reject H0.

  36. Section 7.1 Summary • Stated a null hypothesis and an alternative hypothesis • Identified type I and type I errors and interpreted the level of significance • Determined whether to use a one-tailed or two-tailed statistical test and found a p-value • Made and interpreted a decision based on the results of a statistical test • Wrote a claim for a hypothesis test

  37. Section 7.2 Hypothesis Testing for the Mean (Large Samples)

  38. Section 7.2 Objectives • Find P-values and use them to test a mean μ • Use P-values for a z-test • Find critical values and rejection regions in a normal distribution • Use rejection regions for a z-test

  39. Using P-values to Make a Decision Decision Rule Based on P-value • To use a P-value to make a conclusion in a hypothesis test, compare the P-value with . • If P , then reject H0. • If P> , then fail to reject H0.

  40. Example: Interpreting a P-value The P-value for a hypothesis test is P = 0.0237. What is your decision if the level of significance is • 0.05? • 0.01? Solution:Because 0.0237 < 0.05, you should reject the null hypothesis. Solution: Because 0.0237 > 0.01, you should fail to reject the null hypothesis.

  41. Finding the P-value After determining the hypothesis test’s standardized test statistic and the test statistic’s corresponding area, do one of the following to find the P-value. • For a left-tailed test, P = (Area in left tail). • For a right-tailed test, P = (Area in right tail). • For a two-tailed test, P = 2(Area in tail of test statistic).

  42. Example: Finding the P-value Find the P-value for a left-tailed hypothesis test with a test statistic of z = -2.23. Decide whether to reject H0 if the level of significance is α = 0.01. Solution:For a left-tailed test, P = (Area in left tail) P = 0.0129 z -2.23 0 Because 0.0129 > 0.01, you should fail to reject H0

  43. Example: Finding the P-value Find the P-value for a two-tailed hypothesis test with a test statistic of z = 2.14. Decide whether to reject H0 if the level of significance is α = 0.05. Solution:For a two-tailed test, P = 2(Area in tail of test statistic) 1 – 0.9838 = 0.0162 P = 2(0.0162) = 0.0324 0.9838 z 0 2.14 Because 0.0324 < 0.05, you should reject H0

  44. Z-Test for a Mean μ • Can be used when the population is normal and  is known, or for any population when the sample size n is at least 30. • The teststatisticis the sample mean • The standardizedteststatisticis z • When n  30, the sample standard deviation s can be substituted for .

  45. Using P-values for a z-Test for Mean μ In Words In Symbols • State the claim mathematically and verbally. Identify the null and alternative hypotheses. • Specify the level of significance. • Determine the standardized test statistic. • Find the area that corresponds to z. State H0 and Ha. Identify . Use Table 4 in Appendix B.

  46. Using P-values for a z-Test for Mean μ In Words In Symbols • Find the P-value. • For a left-tailed test, P = (Area in left tail). • For a right-tailed test, P = (Area in right tail). • For a two-tailed test, P = 2(Area in tail of test statistic). • Make a decision to reject or fail to reject the null hypothesis. • Interpret the decision in the context of the original claim. Reject H0 if P-value is less than or equal to . Otherwise, fail to reject H0.

  47. Example: Hypothesis Testing Using P-values In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random selection of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Is there enough evidence to support the claim at = 0.01? Use a P-value.

  48. μ ≥ 30 min μ < 30 min Solution: Hypothesis Testing Using P-values • H0: • Ha: •  = • Test Statistic: • P-value 0.0051 0.01 z -2.57 0 • Decision: 0.0051 < 0.01 Reject H0 At the 1% level of significance, you have sufficient evidence to conclude the mean delivery time is less than 30 minutes.

  49. Example: Hypothesis Testing Using P-values You think that the average franchise investment information shown in the graph is incorrect, so you randomly select 30 franchises and determine the necessary investment for each. The sample mean investment is $135,000 with astandard deviation of $30,000. Is there enough evidence to support your claim at  = 0.05? Use a P-value.

  50. μ = $143,260 μ ≠ $143,260 Solution: Hypothesis Testing Using P-values • H0: • Ha: •  = • Test Statistic: • P-value P = 2(0.0655) = 0.1310 0.05 0.0655 z -1.51 0 • Decision: 0.1310 > 0.05 Fail to reject H0 At the 5% level of significance, there is not sufficient evidence to conclude the mean franchise investment is different from $143,260.

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