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1.2 - the 7 step process of statistical hypothesis testing.
We will cover the seven steps one by one.
Step 1: State the Null Hypothesis
The null hypothesis can be thought of as the opposite of the "guess" the researchers made. In the example presented in the previous section, the biologist "guesses" plant height will be different for the various fertilizers. So the null hypothesis would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as:
\(H_0 \colon \mu_1 = \mu_2 = ⋯ = \mu_T\)
for T levels of an experimental treatment.
Step 2: State the Alternative Hypothesis
\(H_A \colon \text{ treatment level means not all equal}\)
The alternative hypothesis is stated in this way so that if the null is rejected, there are many alternative possibilities.
For example, \(\mu_1\ne \mu_2 = ⋯ = \mu_T\) is one possibility, as is \(\mu_1=\mu_2\ne\mu_3= ⋯ =\mu_T\). Many people make the mistake of stating the alternative hypothesis as \(\mu_1\ne\mu_2\ne⋯\ne\mu_T\) which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. A simple way of thinking about this is that at least one mean is different from all others. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, a possible outcome would be that fertilizer 1 results in plants that are exceptionally tall, but fertilizers 2, 3, and the control group may not differ from one another.
Step 3: Set \(\alpha\)
If we look at what can happen in a hypothesis test, we can construct the following contingency table:
| Decision | In Reality | |
|---|---|---|
| \(H_0\) is TRUE | \(H_0\) is FALSE | |
| Accept \(H_0\) | correct | Type II Error \(\beta\) = probability of Type II Error |
| Reject \(H_0\) | Type I Error | correct |
You should be familiar with Type I and Type II errors from your introductory courses. It is important to note that we want to set \(\alpha\) before the experiment ( a-priori ) because the Type I error is the more grievous error to make. The typical value of \(\alpha\) is 0.05, establishing a 95% confidence level. For this course, we will assume \(\alpha\) =0.05, unless stated otherwise.
Step 4: Collect Data
Remember the importance of recognizing whether data is collected through an experimental design or observational study.
Step 5: Calculate a test statistic
For categorical treatment level means, we use an F- statistic, named after R.A. Fisher. We will explore the mechanics of computing the F- statistic beginning in Lesson 2. The F- value we get from the data is labeled \(F_{\text{calculated}}\).
Step 6: Construct Acceptance / Rejection regions
As with all other test statistics, a threshold (critical) value of F is established. This F- value can be obtained from statistical tables or software and is referred to as \(F_{\text{critical}}\) or \(F_\alpha\). As a reminder, this critical value is the minimum value of the test statistic (in this case \(F_{\text{calculated}}\)) for us to reject the null.
The F- distribution, \(F_\alpha\), and the location of acceptance/rejection regions are shown in the graph below:
Step 7: Based on Steps 5 and 6, draw a conclusion about \(H_0\)
If \(F_{\text{calculated}}\) is larger than \(F_\alpha\), then you are in the rejection region and you can reject the null hypothesis with \(\left(1-\alpha \right)\) level of confidence.
Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting an \(F_{\text{calculated}}\) even greater than what you observe assuming the null hypothesis is true. If by chance, the \(F_{\text{calculated}} = F_\alpha\), then the p -value would be exactly equal to \(\alpha\). With larger \(F_{\text{calculated}}\) values, we move further into the rejection region and the p- value becomes less than \(\alpha\). So, the decision rule is as follows:
If the p- value obtained from the ANOVA is less than \(\alpha\), then reject \(H_0\) in favor of \(H_A\).
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The goal of hypothesis testing is to determine the likelihood that a population parameter, such as the mean, is likely to be true. In this section, we describe the four steps of hypothesis testing that were briefly introduced in Section 8.1: Step 1: State the hypotheses. Step 2: Set the criteria for a decision. Step 3: Compute the test ...
4. HYPOTHESIS TESTING. STEPS IN HYPOTHESIS TESTING. Step 1: State the Hypotheses. Null Hypothesis (H. 0. in the general population there is no change, no difference, or no relationship; the independent variable will have no effect on the dependent variable. o Example. •All dogs have four legs.
Step 1. Determine the null and alternative hypotheses. Null hypothesis: There is no clear winning opinion on this issue; the proportions who would answer yes or no are each 0.50. Alternative hypothesis: Fewer than 0.50, or 50%, of the population would answer yes to this question. The majority do not think Clinton
Review: steps in hypothesis testing about the mean 1.Hypothesis a value ( 0) and set up H 0 and H 1 2.Take a random sample of size n and calculate summary statistics (e.g., sample mean and sample variance) 3.Determine whether it is likely or unlikely that the sample, or one even more extreme, came from a population with mean
the value specified by H0 is called a two-sided (or two-tailed) test, e.g. H0: µ = 100 HA: µ <> 100 I. Whether you use a 1-tailed or 2-tailed test depends on the nature of the problem. Usually we use a 2-tailed test. A 1-tailed test typically requires a little more theory. Introduction to Hypothesis Testing - Page 1
These are the conditions that need to be met in order for the hypothesis test to be performed. If the conditions are not met, then the results of the test are not valid. 4. Calculate the Test Statistic The test statistic varies depending on the test performed, see statistical tests handouts for details. 5. Calculate the P-value
no reason to doubt that the null hypothesis is true. Similarly, if the observed data is "inconsistent" with the null hypothesis (in our example, this means that the sam-ple mean falls outside the interval (90.2, 109.8)), then either a rare event has occurred (rareness is judged by thresholds 0.05 or 0.01) and the null hypothesis is true,
Chapter 5 Hypothesis Testing. Chapter 5Hypothesis TestingA second type of statistical inf. rence is hypothesis testing. Here, rather than use ei-ther a point (or interval) estimate from a random sample to approximate a population parameter, hypothesis testing uses point estimate to decide which of two hypotheses (guesses.
8.3 Hypothesis Testing Summary •Step 1: State hypotheses and select alpha level •Step 2: Locate the critical region •Step 3: Collect data; compute the test statistic •Step 4: Make a probability-based decision about H 0: Reject H 0 if the test statistic is unlikely when H 0 is true—called a "significant"
Hypothesis Testing is one of the real strengths of Statistics. HT's entail our following six steps (p.173): Giving the Null Hypothesis (H0) Giving the Alternative Hypothesis (HA) Setting the level of significance (α) or Decision Rule*. Calculating the Test Statistic (TS) Finding the p-value.
Effect size. Significance tests inform us about the likelihood of a meaningful difference between groups, but they don't always tell us the magnitude of that difference. Because any difference will become "significant" with an arbitrarily large sample, it's important to quantify the effect size that you observe.
Hypothesis testing will rely extensively on the idea that, having a pdf, one can compute the probability of all the corresponding events. Make sure you understand this point before going ahead. We have seen that the pdf of a random variable synthesizes all the probabilities of realization of the underlying events.
o the sampling distribution un. r 0.The hypothesis testing recipeThe basic id. is:If the true parameter was 0...then T (Y) should look like it c. e from f(Y j 0).We compare the observed T (Y) to the sampling distribution under 0.If the observed T (Y) is unlik. ly under the sampling distribution given 0, we reject the null hy.
0. 1. Left-tailed Test. H0 : μ = k H1 : μ < k P-value = P (z < zø) x This is the probability of getting a test statistic as low as or lower than zø x. If P-value ↵, we reject H0 and say the data are statistically significant at the level ↵. If P-value > ↵, we do not reject H0.
The testing of a statistical hypothesis is the application of an explicit set of rules for deciding whether to accept the hypothesis or to reject it. The method of conducting any statistical hypothesis testing can be outlined in six steps : 1. Decide on the null hypothesis H0 The null hypothesis generally expresses the idea of no difference. The
Step 1: State the Null Hypothesis. The null hypothesis can be thought of as the opposite of the "guess" the researchers made: in this example, the biologist thinks the plant height will be different for the fertilizers. So the null would be that there will be no difference among the groups of plants. Specifically, in more statistical language ...
Step 1: The hypothesis statement is H0: μ = 150 versus H1: μ ≠ 150. Observe that μ represents the true-but-unknown mean for the new Krisp-o-Matic machine. The comparison value 150 is the numerical claim, and we want to compare μ to 150. It might seem that the whole problem was set up with H1: μ < 150 in mind.
Hypothesis Tests: Single-Sample tTests. Hypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard deviation. Degrees of Freedom: The number of scores that are free to vary when estimating a population parameter from a sample df = N. 1 (for a Single-Sample.
Step 1: Identify the claim and express in symbolic form. The claim is that the actual mean time Virginians spend on their cell phones per day is greater than 180 minutes, written symbolically as, The greater than symbol means this is a right-tailed test. μ > 180. Step 2: Write the null and alternative hypothesis.
Steps in Hypothesis testing. Statement of hypothesis. Identification of the test statistic and its distribution. Specification of the significance level. Statement of the decision rule. Collection of the data and performance of the calculations. Making the statistical decision. Drawing a conclusion.
Step 1: State the Null Hypothesis. The null hypothesis can be thought of as the opposite of the "guess" the researchers made. In the example presented in the previous section, the biologist "guesses" plant height will be different for the various fertilizers. So the null hypothesis would be that there will be no difference among the groups of ...
Case1: Population is normally or approximately normally distributed with known or unknown variance (sample size n may be small or large), Case 2: Population is not normal with known or unknown variance (n is large i.e. n≥30). 3.Hypothesis: we have three cases. Case I : H0: μ=μ0 HA: μ μ0. e.g. we want to test that the population mean is ...
Step 6 : Use the test statistic to make a decision When we compare the result of step 5 to the decision rule in step 4, it is obvious that 3.024 is greater than the t-critical value of 2.447, and so we reject the null hypothesis. In other words, the mean value of 274 mg/kg is significantly different from the certified value of 250 mg/kg.