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Outline:
Introduction
The study of statistical inference began in the preceding two blogs. It explained how to select a random sample and estimate the value of a population parameter using that sample. For example, we took a sample of five employees from Jones Sprockets, calculated the mean years of service for each sampled employee, and used the sample mean to estimate the mean years of service for all employees. In other words, we used a sample statistic to estimate a population parameter. It continued the study of statistical inference in the previous blog by establishing a confidence interval.
First, let's define what a statistical hypothesis and statistical hypothesis testing are. Following that, we'll walk you through the steps of statistical hypothesis testing. Then, for means and proportions, we do hypothesis tests.
What is Hypthesis Testing?
An educated assumption about something in the world around you is referred to as a hypothesis. It should be able to be put to the test, either through experiment or observation. It is a statistician's act of putting a hypothesis about a population parameter to the test. The analyst's approach is determined by the type of the data and the purpose of the study. Using sample data, this is used to assess the plausibility of a hypothesis. Such information could originate from a wider population or a data-gathering mechanism. In the following descriptions, the word "population" will be used to describe both of these scenarios.
The phrases hypothesis testing and hypothesis testing are frequently used. A declaration, or assumption, about a population parameter, such as the population means, is the starting point for hypothesis testing. A method for determining if a hypothesis is a plausible statement based on sample evidence and probability theory. This assertion is referred to as a hypothesis, as previously stated.
Example:
One hypothesis would be that sales employees in retail electronics stores like Circuit City get a monthly commission of $2,000 on average. We won't be able to contact all of these salespeople to confirm that the mean is, in fact, $2,000. Finding and interviewing every electronics sales associate in the United States would be prohibitively expensive. We must select a sample from the population of all electronics sales associates, generate sample statistics, and accept or reject the hypothesis based on specified decision procedures to evaluate the validity of the assumption (= $2,000). The hypothesis would almost surely be rejected if the sample mean for the electronics sales employees was $1,000. Assume, however, that the sample mean is $1,995. Is that close enough to $2,000 to accept $2,000 as the population mean? Is the $5 difference between the two means due to sampling error, or does it represent a statistically significant difference?
Steps in Hypothesis Testing
Hypothesis testing is organized into a five-step process, and when we reach step 5, we are ready to reject or not reject the hypothesis. However, in the same way that a mathematician "proves" a statement, statisticians' hypothesis testing does not provide proof that something is true. In the same way as the judicial system works, it provides "evidence beyond a reasonable doubt." As a result, there are precise evidentiary guidelines, or procedures, that must be followed.
Here are the steps in Testing a Hypothesis:
Step 1. State the Null Hypothesis and the Alternate Hypothesis
Step 2. Select a Level of Significance
Step 3: Select the Test Statistic
Step 4: Formulate the Decision Rule
Step 1: State the Null Hypothesis (Ho) and the Alternate Hypothesis (H1 or Ha)
The first step is to state the null hypothesis that will be tested and research hypothesis. The null hypothesis is abbreviated as Ho and reads "H sub zero." Hypothesis is represented by the capital letter H, while the subscript zero denotes "no difference." The null hypothesis frequently includes a "not" or "no" phrase, indicating that "no change" has occurred.
The null hypothesis, for example, is that the average number of miles driven on a steel-belted tire does not differ from 60,000. Ho: = 60,000 would be written as the null hypothesis. In general, the null hypothesis is created in order to test anything. The null hypothesis is either rejected or not rejected. The null hypothesis is a proposition that will not be rejected unless our sample data show that it is false.
It's also worth noting that we frequently start the null hypothesis with phrases like "There is no significant difference between..." or "The mean impact strength of the glass is not significantly different from.... “There is no link between...,” says the author. The sample statistic is frequently numerically different from the hypothesized population parameter when we select a sample from a population.
Alternative Hypothesis: If you reject the null hypothesis, you'll come up with an alternate hypothesis. It's abbreviated H1 and reads "H sub one." It's also known as the research hypothesis. If the sample data give adequate statistical evidence that the null hypothesis is untrue, the alternate hypothesis is accepted.
Step 2: Select a Level of Significance
The next stage is to choose a degree of significance after you've established the null and alternative hypotheses. The Greek letter alpha α is used to signify the level of significance. It's also referred to as the risk level. This might be a better term because it refers to the risk of rejecting the null hypothesis when it is actually correct. There is no universally accepted degree of relevance for all tests. The 0.05 level (commonly referred to as the 5% level), the .01 level, the .10 level, or any other level between 0 and 1 is chosen.
Step 3: Select the Test Statistic
Test statistic is a value, determined from sample information, used to determine whether to reject the null hypothesis. There are many test statistics. In this chapter, we use both z and t as the test statistic. In other chapters we will use such test statistics as F and x2, called chi-square. In hypothesis testing for the mean (μ) when a is known or the sample size is large, the test statistic z otherwise the t-test is appropriate
The z value is based on the sampling distribution of X bar, which follows the normal distribution when the sample is reasonably large with a mean (μsubxbar) equal to μ, and a standard deviation σsubxbar, which is equal to σ/√n. We can thus determine whether the difference between Xbar and μ is statistically significant by finding the number of standard deviations Xbar is from μ. CRITICAL VALUE is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.
Step 4: Formulate the Decision Rule
A decision rule specifies the conditions under which the null hypothesis is rejected as well as those under which it is not rejected. The region of rejection, also known as the area of rejection, is the location of all those values that are so large or tiny that their occurrence under a true null hypothesis is extremely unlikely.
In another way, we decide on the null hypothesis based on the value of the test statistic. The selection is made based on the likelihood of achieving a sample mean if the null hypothesis value is true. If the likelihood of obtaining a sample mean is less than 5% when the null hypothesis is true, the null hypothesis should be rejected. If the likelihood of achieving a sample mean when the null hypothesis is true is more than 5%, the null hypothesis should be retained.
Step 5: Make a Decision /Conclusion
Computing the test statistic, comparing it to the critical value, and deciding whether or not to reject the null hypothesis is the fifth and last stage in hypothesis testing. The null hypothesis would not be rejected if the computed test statistic was less than 1.65, say 0.71. It would seem reasonable to attribute such a low computed value to chance, i.e. sampling error. "Do not reject Ho," "We fail to reject Ho," or "The sample data do not enable us to reject Ho" are the terms the researchers like to use.
After completing the statistical analysis and deciding whether to reject or not reject the Null hypothesis, the claim's conclusion must be stated. We need to remember which hypothesis was the claim in order to get the correct phrase. If the claim was the null, your conclusion will be whether there was enough evidence to reject it. Remember that we'll never be able to show the null is false, but failing to reject it is the next best thing. As a result, saying "Accept the Null" is incorrect. If the claim represents an alternative hypothesis, you can decide whether there was enough evidence to support (prove) the alternative hypothesis. Use the table below to assist you get to a good decision. The easiest method to express the conclusion is to provide the test's significance level as well as some information about the claim.
For example, if the alternative claim was that the mean test score was larger than 80, and you chose to Reject the Null, you could conclude: “At the 5% significance level, there is sufficient evidence to support the claim that the mean test score was greater than 80.” The significance level should be included since the decision, and hence the conclusion, could be different if the significance level was not 5%.
"I'm sure I don't have all of the answers or information about Test of Hypothesis here." I'm hoping you'll share your thoughts with Hypothesis testing in the comments area. In the comments, I'd love to hear your thoughts on this.” You can follow to this blog to receive notifications of new posts.
References:
https://www.investopedia.com/terms/h/hypothesistesting.asp https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing https://www.ahajournals.org/doi/full/10.1161/circulationaha.105.586461 https://www.drdawnwright.com/how-to-state-the-conclusion-about-a-hypothesis-test/
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