Basic Steps for Testing a Hypothesis about a Parameter State the population(s) and corresponding parameter(s) of interest. Remember inference is only valid if the sample(s) is representative of the population(s) of interest. State the competing theoriesthat is, the null and alternative hypotheses. The null hypothesis gives a specific value for the parameter, called the hypothesized value or null value. State the significance level for the test. This level should always be set in advance of examining the results. Collect and examine the data and assess if the assumptions are valid. If assumptions are not reasonable, there may be alternative procedures, some of which are discussed in Chapter 15. Compute a test statistic using the data and determine the p- value. The test statistic is a measure of the distance between the sample statistic or point estimate of the parameter and the hypothesized value or null value for the parameter.
Basic Form of a Confidence Interval for a Parameter Point Estimate (a few)(Standard Error of the Point Estimate) The "a few" is either a z* or a t* percentile, which depends on the confidence level desired and the sample size. The confidence level describes how often the procedure (if repeated) provides an interval that actually contains the population parameter.
Lets Do It LDI 11.1
Comparing Two Means Always start off with side-by-side boxplots. Lets look at your Dominate Non-dominate reaction time data. Do the medians appear to line up? Recall if the distribution is symmetric the mean and the median are about the same.
How to determine if Dependent or Independent samples If the data is matched, as in before/after data, then the data are dependent on each other. An example would be weight of a subject prior to starting a new training program versus weight after 6 weeks of the program. If the data come from two distinct populations, say men in new program, compared to woman in new program, the data are independent.
Lets Do It LDI 11.2 Lets run a paired t-test (use the T-test in TI) for the dominate/nondominate data.
Hypothesis Tests Test Statistic for Two Dependent Samples Note:
Claims for Paired Samples There is no difference... Symbolic form: There is a difference... is interpreted as it is expected that the mean of the d values to be different from 0. Symbolic form:
Lets Do It Example 11.3 LDI 11.5 (use data given in class)
Inferences about Two Independent Means Assumptions 1. The two samples are independent. 2. The two samples are randomly selected from normally distributed populations. Less crucial if sample sizes are above 30 and the sample sizes are the same (n 1 = n 2 )
Method Well Use Use your calculator and do not pool the standard deviation. By pooling the standard deviation we are inferring the populations have equal standard deviations. This is tough to test and does not significantly improve the results of the test.
Lets Do It LDI: 11.6 Assume that the population variances are not equal. EX Confidence intervals are found using the 2-SampTint. Give a 95% CI for the Sheep data.
Summary If the data are paired (dependent), find the differences and run a good old T-test. If the data are independent, run a 2-sampTtest assuming that we should not pool the standard deviations.