# Mathematically margin of error and width of

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Mathematically: Margin of Error and Width of Confidence Interval: We can find out the sample size needed, if we are given the margin of error. n = ( ) 2 ( s ¿ 2 ( E ) 2 Keller, Chapter 11-1 Concepts of Hypothesis Testing Keller, Chapter 11-2 Testing the Population Mean Hypothesis Testing : The basic idea behind hypothesis testing is quite similar to confidence intervals except that we pose our question a little bit differently. With hypothesis testing, we are trying to see if a particular claim is valid or not. The basic framework of hypothesis testing is to break every question down into one of two categories. A null hypothesis and an alternative hypothesis. Null Hypothesis : The null hypothesis is the claim being made and what we are seeking to test. Usually we write: H 0 : Here we write whatever the null hypothesis would be. (i.e., H 0 : μ = 350) Alternative or Research Hypothesis : The alternative hypothesis is very simple in most circumstances. It is basically just that the null hypothesis is false. For the most part, we do two-sided hypothesis testing, which means we don’t really care about how the null hypothesis is wrong (too large or too small) just that it is wrong. Usually we write: H 1 : The null hypothesis is false. (i.e., H 1 : μ ≠ 350)
13 After gathering statistical evidence, we will have two possible conclusions. 1. We may conclude “ Reject the null hypothesis and accept the alternative hypothesis ”. This will occur if, and only if, we find strong statistical evidence given our chosen confidence level that the null hypothesis is false. 2. The other possibility is that we will “ Fail to reject the null hypothesis ”. This occurs whenever we don’t find sufficiently strong evidence to reject the null. An important point to note is that this does not mean that we necessarily believe the null is true. Just that we haven’t found sufficient evidence to throw it out. Basically, if we observe something that would be very unlikely if the null were true, then we reject the null. Type I Error : Occurs when we reject a true null hypothesis. The probability of a type I is denoted by α which is also called the significance level. Type II Error : Occurs when a false null hypothesis is not rejected. The probability of a Type II error is denoted by β. Significance Level : The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. Standardized Test Statistic : Informally = Compute a test statistic (often called a t-stat) and see how it compares to the critical values at your significance level. A critical value is a cutoff value that defines the boundaries beyond which less than 5% of sample means can be obtained if the null hypothesis is true. Sample means obtained beyond a critical value will result in a decision to reject the null hypothesis. In a nondirectional two-tailed test, we divide the α value in half so that an equal proportion of area is placed in the upper and lower tail. For example, α = 0 . 05, so we split this probability in half, and get 0.025 in each tail.