9 Hypothesis Tests PDF - Lecture notes Week 10 PDF

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Lecturer: Dr Ana Fernandes...

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9 Hypothesis Tests Formulating Null and Alternative Hypotheses • Hypothesis Testing: Used to determine whether a statement about the value of a population parameter should or should not be rejected. • Null Hypothesis (H0): A tentative assumption about a population parameter. • Alternative Hypothesis (H1): The opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test is establishing. • Testing Research Hypotheses • The research hypothesis should be expressed as the alternative hypothesis. • The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. Type I and Type II Errors • Ideally, we want low probability of type II error (powerful tests) and of type I error (high statistical significance). However, as Pr(type I) falls, Pr(type II) rises. • • In our hypothesis test we control the probability of (the most serious) type I error occurring (α). • α is called the significance level of the test (typically a = 5%) Type I • Hypothesis tests are based on sample data, hence need to allow for the possibility of errors. • Type I error is rejecting H0 when it is true. • The probability of making a Type I error when H0 is true as an equality is called the level of significance. Applications of hypothesis testing that only control the Type I error are often called • significance tests. Type II • A Type II error is accepting H0 when it is false. • A cautious approach is to use the term “do not reject H0” rather than “accept H0”. One Tailed Hypothesis Testing Critical Value Approach • Use the standard normal probability distribution table to find the z-value with an area of α in the lower/upper tail of the distribution. • That value establishes the boundary of the rejection region and is the critical value for the test. Rejection Rule

Population Mean: σ Known Lower-Tailed Test Critical Value Approach

Upper Tailed Test Critical Value Approach

p-Value Approach • p-value is the probability of obtaining the observed, or more extreme, results when H0 is true. The smaller the p-value, the more evidence there is against H0. • • The p-value is used to determine if the null hypothesis should be rejected. Lower Tailed Test p-Value Approach

Upper Tailed Test p-Value Approach

Steps of Hypothesis Testing 1. Formulate the null and alternative hypotheses. 2. Specify the level of significance α. 3. Compute the test statistic from the sample data. p-Value Approach 4. Use the value of the test statistic to compute the p-value. 5. Reject H0 if p-value ≤ α. Critical Value Approach 4. Use the level of significance to determine the critical value and the rejection rule. 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0. Two-Tailed Hypothesis Testing Critical Value Approach • Critical values in both the lower and upper tails of the standard normal curve. • Use the standard normal probability distribution table to find zα/2 (the z-value with an area of α/2 in the upper tail of the distribution) •

• The rejection rule is:

Two Tailed Tests about a Population Mean: σ Known

Confidence Interval Approach • Select a simple random sample from the population and use the value of the sample mean X_ to construct the confidence interval for the population mean µ. • If the confidence interval contains the hypothesised value µ0, do not reject H0. • Otherwise, reject H0. Population Mean: σ Unknown • The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test. However, we can still use the t distribution table to • identify a range for the p-value. • An advantage of computer software packages is that the computer output will provide the p-value for the t distribution.

Population Proportion • Hypothesis test about the value of a population proportion π takes one of these forms: (π0 is the hypothesized value of the population proportion) • Tests about a Population Proportion Test Statistic

Hypothesis Testing & Decision Making • In many decision making situations, the decision maker may want to take action with both the conclusion to reject or not reject H0. • In these situations, it is recommended that the hypothesis testing procedure be extended to include consideration of making a Type II error. Calculating the Probability of Type II Errors Calculating the Probability of a Type II Error in Hypothesis Tests About a Population Mean 1. Formulate the null and alternative hypotheses. 2. Using the critical value approach, use the level of significance α to determine the critical value and the rejection rule for the test. 3. Using the rejection rule, solve for the value of the sample mean corresponding to the critical value of the test statistic. 4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of H0 - this defines the acceptance region. 5. Using the sampling distribution of for a value of µ satisfying the alternative hypothesis, and the acceptance region from step 4, compute the probability that the sample mean will be in the acceptance region. (This is the probability of making a Type II error at the chosen level of µ). Power of the Test • The probability of correctly rejecting H0 when it is false. • For any particular value of µ, the power is 1 – β. • We can show graphically the power associated with each value of µ; such a graph is called a power curve....