Statistics - Principles of Testing | Statistics

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Study GuideStatistics–Principles of Testing1.Quiz: Stating Hypotheses1. QuestionThe null and alternative hypotheses are written about:Answer Choices• a population parameter.• sample data.• a sample statistic.Correct Answera population parameter.Why This Is CorrectHypotheses are statements about thepopulation, such as the population mean(μ)or populationproportion(p).They arenotwritten directly about sample data or sample statistics.2. QuestionWhich hypothesis should be written as an inequality?Answer Choices• the alternative hypothesis• the null hypothesis• either the alternative or the null hypothesis

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Study GuideCorrect Answerthe alternative hypothesisWhy This Is CorrectThenull hypothesis (H₀)is usually written with anequals sign (=).Thealternative hypothesis (Hₐ)is written using aninequalitysuch as:<, >, or ≠3. QuestionWhich of these is a correct null hypothesis?Answer Choices•H₀: x̄ = 12•H₀:μ = 12•H₀:μ > 12Correct AnswerH₀:μ = 12Why This Is CorrectA correct null hypothesis must includeequality (=).SoH₀:μ = 12is correct, whileH₀:μ > 12is not written correctly for a null hypothesis.4. QuestionWhich of these isNOTa correct null hypothesis?Answer Choices•H₀:μ₁= μ₂•H₀:μ₁− μ₂= 0•H₀:μ₁< μ₂

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Study GuideCorrect AnswerH₀:μ₁< μ₂Why This Is CorrectThe null hypothesis must includeequality( = , ≤ , or ≥ ).SinceH₀:μ₁< μ₂has only “<” and no equality, it isnota correct null hypothesis.5. QuestionWhich of these isNOTa correct alternative hypothesis to correspond withH₀:μ = 8?Answer Choices•Hₐ:μ ≠ 8•Hₐ:μ ≤ 8•Hₐ:μ > 8Correct AnswerHₐ:μ ≤ 8Why This Is CorrectIf the null hypothesis isH₀:μ = 8,then the alternative must be one of these forms:Hₐ:μ ≠ 8(two-tailed)Hₐ:μ > 8(right-tailed)Hₐ:μ < 8(left-tailed)ButHₐ:μ ≤ 8includes equality, which isnot allowedin the alternative hypothesis.2.The Test Statistic (Hypothesis Testing Made Simple)Hypothesis testing is a process used in statistics to decide whether an unusual result happenedjustby chance, or whether it happened because somethingrealis going on.

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Study GuideTo do this, we use probability distributions with known areas (especially thenormal distribution) tofind the probability of getting a certain value.Why do we use probability in hypothesis testing?In a hypothesis test, researchers usually hope the probability of getting the resultby chancewill bevery small.Because if that probability islow, it suggests:The result is probably not a coincidenceThe researcher’s explanation (or theory) may be correctQuick real-life exampleImagine you buy a box of raisin cereal and findonly five raisinsinside.You might wonder:•Was itbad luck(random chance)?•Or is there a real problem likefaulty packaging equipment?Hypothesis testing helps answer questions like that.Only Two Outcomes Are PossibleIn hypothesis testing, onlytwo outcomescan happen:1.Reject the null hypothesis2.Do not reject the null hypothesisThat’s it—no other final result is possible.What is a Test Statistic?Atest statisticis a number calculated from sample data that helps us decide whether to reject thenull hypothesis.One common test statistic is thez-score, especially when the population is normally distributed.

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Study GuideYou already know:Values can be converted intoz-scoresThen probabilities can be found using thestandard normal tableCritical Value (Cutoff Point)Before doing the test, we must choose a cutoff value called the:Critical value(ortabled value)This critical value tells us:How extreme a result must be before we reject the null hypothesisIn other words:•If our test statistic gives a probabilitysmallerthan the critical probabilitywe reject the null hypothesis•If not,we do not reject the null hypothesisExample: Sunlight and Depression (Hypothesis Test)Suppose a researcher believes:Sunlight helps prevent depression.So the researcher forms a hypothesis:Hospitals insunny areasshould havelower depression admission ratesthan the nationalaverage.We are given:National annual admission rate =17 per 10,000The researcher takes a sample of hospitals in sunny regions and compares their mean rate to 17.

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Study GuideWriting the HypothesesAlternative (Research) Hypothesis“The mean annual admission rate for depression in sunny areas is less than 17 per 10,000.”Null Hypothesis“The mean annual admission rate for depression in sunny areas is equal to 17 per 10,000.”Notice:•Thealternative hypothesisuses<•Thenull hypothesisalways has=Choosing the Probability Level (Significance Level)Now the researcher must decide:How low should the sample mean be before we say it is “too unlikely” to happen by chance?The researcher chooses:Probability level =5%That means:α = 0.05So the decision rule becomes:If the chance of getting such a low mean isless than 5%, reject the null hypothesis.Finding the Critical z-ScoreNext, we use the standard normal table to find thecritical z-score.

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Study GuideImportant detail:The standard normal table gives the probability of obtaining a valueat z or lower(meaning, the areabelow z)Since we are testing if the mean isless than 17, we care about thelower tailof the curve.So we look up probability:0.05This gives the critical z-score:z =-1.65That means:•Any test statisticbelow −1.65falls into the rejection regionWhat if the hypothesis was “greater than”?If the alternative hypothesis was:Ha:μ1>17Then we would be looking at theupper tail.So we would use probability:0.95which gives:z = 1.65Region of Rejection vs Region of AcceptanceThe critical z-score divides the curve into two parts:Region of rejectionRegion of acceptance(also called “fail to reject” region)

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Study GuideIf the computed test statistic is:•belowthe critical value → reject (H0)•abovethe critical value → do not reject (H0)Figure 1. Thez‐score defines the boundary of the zones of rejection and acceptance.•rejection region on the left•acceptance region•critical value around−1.65•labels like 13 and 17 per 10,000Final Decision ExampleSuppose the sample mean admission rate is:13 per 10,000And the z-score for this sample mean is:z =-1.20Now compare it to the critical value:

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Study Guide•Critical z =−1.65•Computed z =−1.20Since:−1.20 is NOT below − 1.65the test statistic falls in theregion of acceptanceSo the conclusion is:DecisionWecannot rejectthe null hypothesis.MeaningThere is agreater than 5% chanceof getting a sample mean this low (13 per 10,000) just by randomsampling.So wedo not have enough evidenceto say that sunny regions have a significantly lower depressionrate than the national average.3.Quiz: The Test Statistic1. QuestionWhich of these is an example of atest statistic?Answer Choices• the sample mean• the population mean• a z-scoreCorrect Answera z-scoreWhy This Is Correct

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Study GuideAtest statisticis a standardized value (likezort) calculated from sample data and used to decidewhether to reject (H0).Az-scoreis a common test statistic.2. QuestionThe null hypothesis (H0:μ= 12)will be rejected in favor of the alternative hypothesis (Ha:μ>12)atthe (α= 0.05)level if the test statisticzis greater than:Answer Choices• 1.28• 1.65• 1.96Correct Answer1.65Why This Is CorrectThis is aright-tailed testbecause (Ha:μ> 12).At (α= 0.05),the critical z-value for a right-tailed test is:z = 1.65So we reject (H0) ifz > 1.65.3. QuestionA test is conducted with the null hypothesis (H0:μ= 10)vs. the alternative hypothesis (Ha:μ< 10)atthe (α= 0.05)level.The test statistic isz =–1.75. The correct conclusion is:Answer Choices• fail to reject the null hypothesis• reject the null hypothesis• only possible at the (α= 0.01)level

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