HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 10.1 Fundamentals of Hypothesis Testing With more helpful content added by D.R.S., University of Cordele Hypothesis testing is a technique for testing a claim about a population parameter using statistical principles.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Null and Alternative Hypotheses The alternative hypothesis, denoted by is a mathematical statement that describes a population parameter, and it is the hypothesis that the researcher is aiming to gather evidence in favor of; it is also referred to as the research hypothesis. The subscript is a lowercase little letter “a”

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Null and Alternative Hypotheses Null and Alternative Hypotheses (cont.) The null hypothesis, denoted by is the mathematical opposite of the alternative hypothesis; it will always include equality. The subscript is the digit zero, 0. This means the Null Hypothesis will always involve the symbol “=“ or “≥” or “≤”. And the Alternative Hypothesis will always involve the symbol “≠” or “ ”.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. What “population parameter” is it? Sometimes it’s a hypothesis test about μ, which is the _____________________ _________ value of some measurement. Sometimes it’s a hypothesis test about p, the ________________ of the population that has some characteristic (a certain medical condition, or owns some product, or prefers some flavor, etc.) Hypothesis testing is a technique for testing a claim about a population parameter using statistical principles.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses An engineer has designed a valve… tested on 180 engines and the mean pressure was 4.8 lbs/sq.in. Assume σ is known; σ =1. If the valve was designed to produce a mean pressure of 4.6 lbs/sq.in., is there sufficient evidence at the 0.05 level that the value does not perform to specifications? H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim)

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 440 gram setting. Is there sufficient evidence at the 0.02 level that the bags are overfilled? Assume the population is normally distributed. H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim) Sometimes it’s a good idea to start with the Alternative Hypothesis, then go back to the Null Hypothesis.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses Using traditional methods it takes 106 hours to receive an advanced license. A new and different training program has been proposed. A researcher believes the new technique may lengthen the training time and examined a sample of 60 students… H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim)

Do I write hypotheses using μ (mu) or p ? If the hypotheses are about a population mean, use μ. If the hypotheses are about a population proportion use p. If the hypotheses are about a population percentage, it is really a proportion problem, so p is the right letter.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses A sample of 900 computer chips revealed that 4.1% of the chips fail in the first 1000 hours of use. The company’s promotional literature says that 4.5% fail in the first 1000 hours of use. Is there sufficient evidence at the 0.02 level to disprove the claim? H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim)

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses Company claims that 99% of its customers are satisfied. A competitor thinks that’s inflated and commissions a secret survey of a random sample of 77 of those customers. H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim) Sometimes it’s a good idea to start with the Alternative Hypothesis, then go back to the Null Hypothesis.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example of Writing Hypotheses A troublemaking busybody excuse of a newspaper published a story saying that only 5 out of 8 graduates of Tick Tock Tech find jobs related to their major field within a year of graduation. T.T.T. takes a random sample of 135 graduates and learns that 93 of them have successful, rewarding, degree-related jobs. H 0 : ____________________ (the default “fact”) H a : ____________________ (the contrary claim)

Where does the “=” live? The Null Hypothesis, H 0, always has the “=” sense: = or ≤ or ≥. The Alternative Hypothesis, H a, always has a strict inequality: ≠ or The possibilities are: H 0 :μ = some value and H a :μ ≠ that value H 0 :μ ≤ some value and H a :μ > that value H 0 :μ ≥ some value and H a :μ < that value And similarly for proportions with p.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. What is a Test Statistic ? And what does “Statistically Significant” mean? Test Statistic A test statistic is the value used to make a decision about the null hypothesis and is derived from the sample statistic. A sample statistic is said to be statistically significant if it is far enough away from the presumed value of the population parameter to conclude that it would be unlikely for the sample statistic to occur by chance if the null hypothesis is true.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Test Statistic and the Level of Significance Test Statistic (cont.) The level of significance, denoted by , is the probability of making the error of rejecting a true null hypothesis in a hypothesis test;  = 1  c. The Level of Significance, α (alpha), represents the small area in the tail(s), That’s where H a “wins”, because we reject H 0. The confidence level, c, represents the big area in which H a “loses”, because we fail to reject H 0, for lack of strong enough evidence.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Test Statistic Conclusions for a Hypothesis Test Reject the null hypothesis. Fail to reject the null hypothesis. Your hypothesis test will always conclude with one of these two endings. Either you have strong enough evidence to reject H 0. Or you don’t have strong enough evidence. We do NOT say we “accept H a ” or “accept H 0 ”. Rather, we say we “fail to reject H 0 ”.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Interpreting the Conclusion to a Hypothesis Test A city official claimed that stationing an officer outside the elementary school just one morning a week was enough to slow the average driver to 28 mph. Some “concerned” parent said the cars were really traveling much faster than that and demanded more action. A group of Criminal Justice students ran a radar speed check on 81 randomly selected vehicles. And some statistics students designed and performed a hypothesis test. Their decision was “fail to reject the null hypothesis.”

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Interpreting the Conclusion to a Hypothesis Test The hypotheses are: H 0 : _______________ and H a : ________________ The conclusion is (a)There is sufficient evidence at the _____ level of significance that drivers are still speeding. (b)There is not sufficient evidence at the _____ level of significance that drivers are still speeding.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Interpreting the Conclusion to a Hypothesis Test Motorhead Monthly Magazine believes that 50% of their readership plans to buy a brand new vehicle within the next twelve months. They hire a consulting firm that decides to use a 0.02 level of significance, chooses a random sample of readers, crunches the data, calculates the test statistic, and determines that the correct decision is to reject the null hypothesis.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Interpreting the Conclusion to a Hypothesis Test The hypotheses are: H 0 : _______________ and H a : ________________ The conclusion is (a)There is sufficient evidence at the 0.02 level of significance that the percentage is not 50%. (b)There is not sufficient evidence at the 0.02 level of significance that the percentage is not 50%.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. How do you do a Hypothesis Test? Performing a Hypothesis Test 1.State the null and alternative hypotheses. 2.Determine which distribution to use (z or t) for the test statistic, and state the level of significance (decide what is the value of alpha, α) 3.Gather data and calculate the necessary sample statistics. 4.Draw a conclusion (reject H 0 or fail to reject H 0 ) and interpret the decision.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Type I Error – the probability of rejecting a true null hypothesis,  Type II Error – the probability of failing to reject a false null hypothesis,  Definitions: Types of Errors The Reality H 0 is trueH 0 is false Your Decision H 0 is rejectedType I errorCorrect Decision H 0 is not rejectedCorrect DecisionType II error Does your Decision agree with Reality ? Consider: what might lead to an error? Who or What is at fault?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.7: Determining the Type of Error A television executive believes that at least 99% of households in the United States have at least one television. An intern at the executive’s company is given the task of using a hypothesis test to determine whether the percentage is actually less than 99%. The hypothesis test is completed, and based on the sample collected, the intern decides to fail to reject the null hypothesis. If, in reality, 96.7% of households own a television set, was an error made? If so, what type?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.7: Determining the Type of Error (cont.) Write hypotheses. Think: “What is the intern gathering data for?” That decides H a, and H 0 follows. H 0 : __________ H a : __________ The Reality H 0 is trueH 0 is false Your Decision H 0 is rejectedType I errorCorrect Decision H 0 is not rejectedCorrect DecisionType II error Which box represents what happened? based on the sample collected, the intern decides to fail to reject the null hypothesis. Suppose that, in reality, 96.7% of households own a television set.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.8: Determining the Type of Error Insurance companies commonly use 1000 miles as the mean number of miles a car is driven per month. One insurance company claims that, due to our more mobile society, the mean is more than 1000 miles per month. The insurance company tests its claim with a hypothesis test and decides to reject the null hypothesis. Assume that in reality, the mean number of miles a car is driven per month is 1250 miles. Was an error made? If so, what type?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.8: Determining the Type of Error (cont.) The decision was to { reject, fail to reject } the null hypothesis. So what happened? The Reality H 0 is trueH 0 is false Your Decision H 0 is rejectedType I errorCorrect Decision H 0 is not rejectedCorrect DecisionType II error Hypotheses: H 0 : ___________ H a : _____________

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.9: Determining the Type of Error A study on the effects of television-viewing on children reports that children watch a mean of 4.0 hours of television per night. Kiko believes that the mean number of hours children in her neighborhood watch television per night is not 4.0. She performs a hypothesis test and rejects the null hypothesis. Assume that in reality, children in her neighborhood do watch a mean of 4.0 hours of television per night. Did she make an error? If so, what type?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.9 Kiko’s TV viewing test The decision was to { reject, fail to reject } the null hypothesis. So what happened? The Reality H 0 is trueH 0 is false Your Decision H 0 is rejectedType I errorCorrect Decision H 0 is not rejectedCorrect DecisionType II error Hypotheses: H 0 : ___________ H a : _____________