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Hypothesis Testing IntroductionWith hypothesis testing, we are making a claim that a sample that is treated differently may or may not yield a different result than the population. Example of Hypothesis TestingFor example: a teacher has been teaching math class the same way for several years. The average test score over those years is 85. For one semester the teacher changes their method of instruction and the students in that cohort receive an average test score of 89. The question we have is if this new method of instruction actually improved the test scores for this cohort or should we consider the test average to be a random deviation. Hypothesis testing proceduresWith hypothesis testing we have to do these things:
In addition there are two tests we can do, the z-test and t-test. There are also three methods of hypothesis testing.
This page addresses the Traditional Method and the following pages address the P-Value Method and Confidence Interval Method. Traditional MethodA statistical hypothesis is a conjecture about a population parameter. The conjecture may or may not be true. Statistical HypothesesThere are two statistical hypotheses:
So in our teaching example from above the null hypothesis \(H_0\) would be that the new method of math instruction makes no difference on the test scores of the students. Our alternative hypothesis, \(H_1\), is that the students difference between the There are also three different situations: Situation A The Two-tailed TestIn this scenario we’re looking for either no difference in a parameter between a population and a sample or a difference between the population and sample. \[H_0: \mu = k \text{ and } H_1: \mu \neq k\] For example an experimental drug is created, does it change a patient’s blood pressure? If \(H_0: \mu = k\) then the drug doesn’t change a patient’s blood pressure. If \(H_1: \mu \neq k\) then the drug does change the patient’s blood pressure. Significantly if \(H_1: \mu \neq k\) it could be higher or lower. Situation B The Right-tailed TestThere’s a new experimental strain of corn that will grow faster. This is a right-tailed test, we’re specifically looking for an increase in the rate of growth so in this case we’re looking for this: \[H_0: \mu = k \text{ and } H_1: \mu > k\] If \(H_0: \mu = k\) then the hypothesis is false and the new strain of corn doesn’t grow faster. If \(H_1: \mu > k\) then the strain of corn does indeed grow faster than existing strains. Situation C The Left-tailed TestIn this case we might have an experiment testing whether, for example, climate change lowers the rate of tree growth in a forest. This is the comparison that we are looking at. \[H_0: \mu = k \text{ and } H_1: \mu < k\] In this case if \(H_0: \mu = k\) then, yep, no change. If \(H_1: \mu < k\) then tree growth declines for the trees in question. A statistical test uses data obtained from a sample to make a decision about whether the null hypothesis should be rejected. The numerical value obtained from the a statistical test is called the test value. ErrorsThe outcome of any statistical test can yield one of four possible outcomes. The test will yield one of two outcomes, that the null hypothesis is true or false, however the test result can be different from what happens in the real world. Because we’re testing a sample and not a population there could be unforseen factors that affect the test result on a sample group or just random chance could yield a result that doesn’t hold up for the full population.
Measuring significance levels
This is symbolized by the letter alpha \(\alpha\). \(P(\text{Type I}) = \alpha\) The letter Beta \(\beta\) is used to symbolize the maximum probability of committing a type II error. \(P(\text{Type II}) = \beta\) The three levels of significance most commonly used are 0.10, 0.05, and 0.01. Critical Value, Critical Region, Non-Critical RegionThe critical region shows the range of values that indicates there is a significant difference and the null hypothesis should be rejected. The non-critical region shows the range of values of the test that indicate that the difference was probably due to chance and that ht enull hypothesis should be rejected. The critical value separates the critical region from the non-critical region. The plot below shows a right-tailed test with a \(P(\text{Type I}) = \alpha = 0.05\). The red area is the critical region, the area with no color is the non-critical region, and the line separating them is the critical value. Finding the Critical ValueThis depends on the type of test. If the test is left-tailed, right-tailed, or two-tailed. Left-tailed:
Pretty simple, use the negative z-score table (below ) to find the score that matches the percentage of values that would indicate \(H_0\) is false. So for example if the critical value you are using that \(H_0\) is false is 0.05, then find 0.05 in the table and then the zscore in the row and column which would be -1.64.
Use the qnorm function we can find the critical values for three different probabilities of getting a Type I error, .1, .05, and .01.
So the three most commonly used probabilities would be: \[ \begin{aligned} H_0: \mu = k \\ H_1: \mu < k \\ \end{aligned} \left \{ \begin{aligned} \alpha = 0.10, C.V. = -1.28 \\ \alpha = 0.05, C.V. = -1.65 \\ \alpha = 0.01, C.V. = -2.33 \end{aligned} \right \} \] Right-tailed:
Pretty simple, use the complementary cumulative z-score table starting from the left to find the score that matches the percentage of values that would indicate \(H_0\) is false. So for example if the probability that \(H_0\) is false is 0.05 for a right-tailed test, then find 0.05 in the table and then the zscore in the row and column which would be 1.64.
Use the qnorm function
So the three most commonly used probabilities would be: \[ \begin{aligned} H_0: \mu = k \\ H_1: \mu > k \\ \end{aligned} \left \{ \begin{aligned} \alpha = 0.10, C.V. = 1.28 \\ \alpha = 0.05, C.V. = 1.65 \\ \alpha = 0.01, C.V. = 2.33 \end{aligned} \right \} \] Two-tailed:
Now we’re looking for two numbers, use the complementary cumulative and the negative z-score tables. Take alpha, divide by two and find the probabilities for both. So for \(\alpha = 0.05\) we’ll look for \(\alpha = 0.05/2 = 0.025\) and 1-0.025 = .975
So the three most commonly used probabilities would be: \[ \begin{aligned} H_0: \mu = k \\ H_1: \mu \neq k \\ \end{aligned} \left \{ \begin{aligned} \alpha = 0.10, C.V. = \pm1.65 \\ \alpha = 0.05, C.V. = \pm1.96 \\ \alpha = 0.01, C.V. = \pm2.58 \end{aligned} \right \} \] Z-Score TablesNegative Z-score Table
Cumulative
Complementary cumulative
Cumulative From The Mean
Is the test value separates the critical region from the noncritical region?The critical value separates the critical region from the noncritical region. The symbol for critical value is C.V. The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected.
Is the value of test statistic that separates the rejection and nonA critical value separates the rejection region from the non-rejection region. 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.
Which value separates the critical region from the?C. The P-value separates the critical region from the values that do not lead to rejection of the null hypothesis.
What does the critical value separate?In a test of hypothesis, a critical value is a number that separates two regions: the critical region, that is, the set of values of the test statistic that lead to a rejection of the null hypothesis; the acceptance region, that is, the set of values for which the null is not rejected.
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