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Statistics - Principles of Testing

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    Study Guide Statistics – Principles of Testing 1. Quiz: Stating Hypotheses 1. Question The null and alternative hypotheses are written about: Answer Choices • a population parameter. • sample data. • a sample statistic. Correct Answer a population parameter. Why This Is Correct Hypotheses are statements about the population , such as the population mean ( μ) or population proportion (p) . They are not written directly about sample data or sample statistics. 2. Question Which 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 Guide Correct Answer the alternative hypothesis Why This Is Correct The null hypothesis (H ₀ ) is usually written with an equals sign (=) . The alternative hypothesis (Hₐ) is written using an inequality such as: <, >, or ≠ 3. Question Which of these is a correct null hypothesis? Answer Choices • H ₀ : x̄ = 12 • H ₀ : μ = 12 • H ₀ : μ > 12 Correct Answer H ₀ : μ = 12 Why This Is Correct A correct null hypothesis must include equality (=) . So H ₀ : μ = 12 is correct, while H ₀ : μ > 12 is not written correctly for a null hypothesis. 4. Question Which of these is NOT a correct null hypothesis? Answer Choices • H ₀ : μ ₁ = μ ₂ • H ₀ : μ ₁ − μ ₂ = 0 • H ₀ : μ ₁ < μ ₂

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    Study Guide Correct Answer H ₀ : μ ₁ < μ ₂ Why This Is Correct The null hypothesis must include equality ( = , ≤ , or ≥ ). Since H ₀ : μ ₁ < μ ₂ has only “<” and no equality, it is not a correct null hypothesis. 5. Question Which of these is NOT a correct alternative hypothesis to correspond with H ₀ : μ = 8 ? Answer Choices • Hₐ: μ ≠ 8 • Hₐ: μ ≤ 8 • Hₐ: μ > 8 Correct Answer Hₐ: μ ≤ 8 Why This Is Correct If the null hypothesis is H ₀ : 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) But Hₐ: μ ≤ 8 includes equality, which is not allowed in 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 happened just by chance , or whether it happened because something real is going on.

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    Study Guide To do this, we use probability distributions with known areas (especially the normal distribution ) to find 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 result by chance will be very small. Because if that probability is low , it suggests: The result is probably not a coincidence The researcher’s explanation (or theory) may be correct Quick real - life example Imagine you buy a box of raisin cereal and find only five raisins inside. You might wonder: • Was it bad luck (random chance)? • Or is there a real problem like faulty packaging equipment ? Hypothesis testing helps answer questions like that. Only Two Outcomes Are Possible In hypothesis testing, only two outcomes can happen: 1. Reject the null hypothesis 2. Do not reject the null hypothesis That’s it — no other final result is possible. What is a Test Statistic? A test statistic is a number calculated from sample data that helps us decide whether to reject the null hypothesis. One common test statistic is the z - score , especially when the population is normally distributed.

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    Study Guide You already know: Values can be converted into z - scores Then probabilities can be found using the standard normal table Critical Value (Cutoff Point) Before doing the test, we must choose a cutoff value called the: Critical value (or tabled value ) This critical value tells us: How extreme a result must be before we reject the null hypothesis In other words: • If our test statistic gives a probability smaller than the critical probability we reject the null hypothesis • If not, we do not reject the null hypothesis Example: Sunlight and Depression (Hypothesis Test) Suppose a researcher believes: Sunlight helps prevent depression. So the researcher forms a hypothesis: Hospitals in sunny areas should have lower depression admission rates than the national average. We are given: National annual admission rate = 17 per 10,000 The researcher takes a sample of hospitals in sunny regions and compares their mean rate to 17.

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    Study Guide Writing the Hypotheses Alternative (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: • The alternative hypothesis uses < • The null hypothesis always 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 = 0.05 So the decision rule becomes: If the chance of getting such a low mean is less than 5% , reject the null hypothesis. Finding the Critical z - Score Next, we use the standard normal table to find the critical z - score .

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    Study Guide Important detail: The standard normal table gives the probability of obtaining a value at z or lower (meaning, the area below z ) Since we are testing if the mean is less than 17 , we care about the lower tail of the curve. So we look up probability: 0.05 This gives the critical z - score: z = - 1.65 That means: • Any test statistic below −1.65 falls into the rejection region What if the hypothesis was “greater than”? If the alternative hypothesis was: H a : μ 1 > 17 Then we would be looking at the upper tail . So we would use probability: 0.95 which gives: z = 1.65 Region of Rejection vs Region of Acceptance The critical z - score divides the curve into two parts: Region of rejection Region of acceptance (also called “fail to reject” region)

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    Study Guide If the computed test statistic is: • below the critical value → reject (H 0 ) • above the critical value → do not reject (H 0 ) Figure 1. The z ‐ 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 10,000 Final Decision Example Suppose the sample mean admission rate is: 13 per 10,000 And the z - score for this sample mean is: z = - 1.20 Now compare it to the critical value:

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    Study Guide • Critical z = − 1.65 • Computed z = − 1.20 Since: − 1.20 is NOT below − 1.65 the test statistic falls in the region of acceptance So the conclusion is: Decision We cannot reject the null hypothesis. Meaning There is a greater than 5% chance of getting a sample mean this low (13 per 10,000) just by random sampling. So we do not have enough evidence to say that sunny regions have a significantly lower depression rate than the national average. 3. Quiz: The Test Statistic 1. Question Which of these is an example of a test statistic ? Answer Choices • the sample mean • the population mean • a z - score Correct Answer a z - score Why This Is Correct

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    Study Guide A test statistic is a standardized value (like z or t ) calculated from sample data and used to decide whether to reject (H 0 ). A z - score is a common test statistic. 2. Question The null hypothesis (H 0 : μ = 12) will be rejected in favor of the alternative hypothesis (H a : μ > 12) at the ( α = 0.05) level if the test statistic z is greater than: Answer Choices • 1.28 • 1.65 • 1.96 Correct Answer 1.65 Why This Is Correct This is a right - tailed test because (H a : μ > 12). At ( α = 0.05), the critical z - value for a right - tailed test is: z = 1.65 So we reject (H 0 ) if z > 1.65 . 3. Question A test is conducted with the null hypothesis ( H 0 : μ = 10) vs. the alternative hypothesis (H a : μ < 10) at the ( α = 0.05) level. The test statistic is z = – 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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