Hypothesis Testing Notes

Hypothesis Testing  Is used to determine whether the difference in two groups is likely to be caused by chance  If you flip a coin 30 times for example, you probably wouldn’t be surprised if it was heads up16 times and tails up 14 times…but you would probably be suspicious if it was heads up 28 times and tails up only 2 times.  Hypothesis testing will allow us to determine whether it is likely or unlikely that there is bias in a given situation.

Hypothesis Testing Continued  We will begin with a null hypothesis, which states that there is no difference between the two groups being tested.  A null hypothesis is often times the opposite of what is expected to happen. We use the null hypothesis so that we can allow the data to contradict it.  In a randomized controlled experiment, the null hypothesis will always be that there is no difference in the value of the variable for the control group and the treatment group.

Let’s look at an example.  A teacher wants to know if her first block class performs better on a quiz than her fourth block class. She compares the scores of 10 randomly chosen students in each class.  First Block: 76, 81, 71, 80, 88, 66, 79, 67, 85, 68  Fourth Block: 80, 91, 74, 92, 80, 80, 88, 67, 75, 78  Her null hypothesis would be: The students in first block will have the same quiz grades as the students in fourth block.

Example continued:  Let’s do a five-number summary for each class:  First Block: min = 66, Q1 = 68, median = 77.5, Q3 = 81, max = 88  Fourth Block: min = 67, Q1 = 75, median = 80, Q3 = 88, max = 92  There is a large difference in the two classes that is unlikely to be caused by chance.  The teacher should reject her null hypothesis, which means that students in her afternoon class do better on quizzes.

 If a sample contains at least 30 individuals, you can use a z-test to be more precise with your hypothesis testing.  To find the z-value of a statistic for a sample with n individuals, a sample mean of x, a population mean of µ, and a standard deviation σ, we do:  A large z-value will tell us to reject the null hypothesis.  If the absolute value of z is greater than 1.96, then you can reject the null hypothesis with 95% certainty.  If the absolute value of z is less than 1.96, then you do no have enough evidence to reject the null hypothesis.

Example  A test prep company says that it can boost SAT scores to an average of In a random sample of 36 students who took the course, the average was 1745 with a standard deviation of 210. Is there enough evidence to reject the claim?

Example continued  Our null hypothesis would be: There is no difference in the sample and the population.  Let’s find the z-value. The mean is 1800, the sample mean is 1745, the standard deviation is 210, and n = 36:  Because the absolute value of z is less than 1.96, you do not have enough evidence to reject the claim with 95% confidence.  This doesn’t mean their claim is true, but we don’t have enough information to prove it false.