1 Lecture note 4 Hypothesis Testing Significant Difference ©

2 In short, here’s what happens. The researcher 1. Develops a hypothesis that states that there is no difference between a population parameter and observed statistic. 2. A criterion is set determining the size of the difference. If the difference is larger than the standard (or expected value), you reject the hypothesis

3 More Significant Difference 3. Determine the actual size of the difference-- mathematically 4. If the difference discovered is beyond the criterion, you reject the hypothesis. If the difference is smaller than the criterion set in, you fail to reject the null hypothesis in #1.

4 example Title: The difference in the achievement in applied statistics between male and female in Palestine University Question: Is there any difference in the achievement in applied statistics between male and female in Palestine University? Hypothesis: Null Hypothesis H0: there is no difference at significant level (α≤ 0.05) in the achievement in applied statistics between male and female in Palestine University. Alternative hypothesis Ha : there is difference at significant level (α≤ 0.05) in the achievement in applied statistics between male and female in Palestine University.

5 Binary Decision Remember: The value being tested is the value in the null hypothesis. Remember: The value being tested is a parameter, a population value The researcher either rejects the null hypothesis or fails to reject the null hypothesis.

6 Step 1. State the Hypothesis The mean SAT score of the freshman class is 455 The alternative hypothesis includes the outcomes not covered by the null hypothesis. The alternative hypothesis is considered to be the research hypothesis. This one says, “The mean SAT score of the freshman class is NOT 455.”

7 Specify the Test Statistic These test statistics refer to a sampling distribution “z” -- normal distribution “t” -- t distribution Chi square –chi square distribution F –F distribution In our case, since we are concerned with the mean of a single population, and the population standard deviation is known, the test statistic is z

8 Errors in Hypothesis Testing Four things can happen: 1. A true hypothesis is rejected 2. A true hypothesis is not rejected 3. A false hypothesis is not rejected. 4. A false hypothesis is rejected Type I Error: is when we reject a true null hypothesis Type II Error is when we do not reject a false null hypothesis

9 Level of Significance Criterion for rejecting the null hypothesis is known as level of significance(alpha level). Alpha level: probability of making a Type I Error when testing the null hypothesis. This level of significance is established before collecting any data. Computer stat packages will compute an exact significance..05 or.01 are the most frequently used levels of significance. The idea is that the decision to reject the null hypothesis could be incorrect 5% or 1% of the time, respectively.

10 Type II Error The probability of not rejecting a false null hypothesis (beta) Increasing the level of significance, decreases the probability of making a Type II Error Lowering the alpha level, increases the probability of making a Type II Error.

11 Region of Rejection Area underneath the curve that equals the level of significance This is the area of the sampling distribution that represents those values of the sample mean that are improbable if the null hypothesis is true. Critical values of the test statistic are those values in the sampling distribution that represent the beginning of the region of rejection.

12 Two-tailed and One tailed tests If the alternative hypothesis uses an equal sign, this indicates a two tailed test(nondirectional). In this case, the region of rejection is located in both tails. If the alternative hypothesis uses a greater or less than sign (<>), this is a directional test. In this case, the region of rejection is located is one tail of the sampling distribution

13 Back to the Criterion Let’s set our criterion for our example at.05. Mu = 455 Sigma = 100 “n”= 144 Standard error of the mean is 8.33 Sample mean = 535

14 Critical Values

15 Compute the Test Statistic

16 Decide 9.60 is 9.6 standard errors above mu. This is far above the z values that begin the region of rejection (-1.96 and +1.96). Since the observed value of the test statistic (+9.60) exceeds the critical value (1.96), the probability is less than.05 that the observed sample mean (X-bar =535) will have occurred by chance if the null hypothesis is true (Mu = 455)