In order to test the hypotheses, we select a random sample of American males in 2006 and measure their weights. Suppose we have resources available to recruit n=100 men into our sample. We weigh each participant and compute summary statistics on the sample data. Suppose in the sample we determine the following:
Do the sample data support the null or research hypothesis? The sample mean of 197.1 is numerically higher than 191. However, is this difference more than would be expected by chance? In hypothesis testing, we assume that the null hypothesis holds until proven otherwise. We therefore need to determine the likelihood of observing a sample mean of 197.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true or under the null hypothesis). We can compute this probability using the Central Limit Theorem. Specifically,
The criterion will let us conclude whether (reject null hypothesis) or not (accept null hypothesis) the treatment (prenatal alcohol) has an effect (on birth weight).
The alternative hypothesis can bedirectional or non-directional.“Eating oatmeal lowers cholesterol” is a directional hypothesis; “Amountof sleep affects test scores” is non-directional.
(Notice that we use the sample standard deviation in computing the Z score. This is generally an appropriate substitution as long as the sample size is large, n > 30. Thus, there is less than a 1% probability of observing a sample mean as large as 197.1 when the true population mean is 191. Do you think that the null hypothesis is likely true? Based on how unlikely it is to observe a sample mean of 197.1 under the null hypothesis (i.e.,
We decide: "The data (and its sample mean) are significantly different than the value of the mean hypothesized under the null hypothesis, at the .01 level of significance." This decision is likely to be wrong (Type I error) 1 time out of 100.
The p-value= .004 indicates that we should decide in favor of the alternative hypothesis. Thus we decide that less than 40% of college women think they are overweight.
The "Z-value" (-2.62) is the test statistic. It is a standardized score for the difference between the sample p and the null hypothesis value p = .40. The p-value is the probability that the z-score would lean toward the alternative hypothesis as much as it does if the true population really was p = .40.
We decide: "The data (and its sample mean) are significantly different than the value of the mean hypothesized under the null hypothesis, at the .001 level of significance." This decision is likely to be wrong (Type I error) 1 time out of 1000.
Alternatively to going through these 5 steps by hand we could have invoked Minitab. If you want to try it, open Minitab and go to Stat > Basic Stat > 1- proportion and click Summarize Data and enter 129 for number of trials and 37 for number of events. Next select the checkbox for Perform Hypothesis Test and enter the hypothesized po value. Finally, the default alternative is "not equal". To select a different alternative click Options and select the proper option from the drop down list next to Alternative, plus click the box for Test and Interval using Normal Approximation. The results of doing so are as follows:
When we get the data we will calculate Z and then look it up in the Z table to see how unusual the obtained sample's mean is, if the null hypothesis Ho is true.
Here's what happens:
The P value is way below .00001, so we reject the null hypothesis that there is an unrestrictive selection process for admitting students to UNC.
The decision rule is a statement that tells under what circumstances to reject the null hypothesis. The decision rule is based on specific values of the test statistic (e.g., reject H0 if Z > 1.645). The decision rule for a specific test depends on 3 factors: the research or alternative hypothesis, the test statistic and the level of significance. Each is discussed below.
The final conclusion is made by comparing the test statistic (which is a summary of the information observed in the sample) to the decision rule. The final conclusion will be either to reject the null hypothesis (because the sample data are very unlikely if the null hypothesis is true) or not to reject the null hypothesis (because the sample data are not very unlikely).