Hypothesis Testing: Two-Sample Case for the Mean
Many cases in the social sciences involve a hypothesis about the difference between two groups (i.e. men and women, control and experiment). We analyze statistics from two samples, and the hypothesis and confidence interval would deal with the difference between two population means. The following factors are important in hypothesis testing:
1. probability theory
2. the sampling distribution of the statistic
3. the errors inherent in hypothesis testing and estimation 4. the level of significance and the level of confidence
5. the directional nature of the alternative hypothesis
1. State the hypotheses.
2. Set the criterion for rejecting H0.
3. Compute the test statistic.
4. Construct the confidence interval.
5. Interpret the results.
Hypothesis of Differences
• There is no difference between mean of group 1 and the mean of group 2. • [pic] or [pic]
o to test this difference, we determine the difference between the statistic (the difference between the means), and the hypothesized value for the parameter (0). o if the population variance is known, the sampling distribution of differences is normally distributed. o if the population variance is UNKNOWN, the sampling distribution of differences is the t distribution, for the appropriate degrees of freedom.
Assumptions that must be considered:
1. Independence. The samples must be independent, that is, the scores of one sample in no way influence the scores of the other sample. a. Random selection from the population, then random assignment to the groups b. Random selection from two populations
2. Homogeneity of Variance. Since a pooled estimate of the sample variance is used, we must assume that the variance in population 1 is equal to the variance in population 2. a. If the two samples are of equal size, this is not a problem b. If the two sample sizes are unequal (radically unequal) then alternative procedures must be use and will be discussed below.
Sampling Distribution of Differences
As the sizes for populations 1 and 2 increase, the sampling distribution of differences between sample means has the following properties:
• Shape: The distribution of differences between sample means approaches a normal distribution • Central tendency: The mean of the distribution of differences of sample means equals [pic]. • Variability: The standard deviation of the distribution of differences of sample means—called the standard error of the difference between means equals: [pic] . This is the estimated standard error of the difference.
The pooled estimate of variance is calculated as follows
Testing the Hypothesis
The basic formula for testing the hypothesis is
The basic formula for constructing a confidence interval is
Interpreting the Results:
“Since the observed value of t(test statistic) exceeds the critical value (critical value), the null hypothesis is rejected in favor of the directional alternative hypothesis. The probability that the observed difference (difference between means) would have occurred by chance, if in fact the null hypothesis is true, is less than .05.
Independent Samples When Variances are Unequal
If variances are unequal and n’s are unequal, adjustments must be made for estimating the standard error of the difference and to the degrees of freedom used in the statistical test.
Calculate an F-ratio: If the variances are equal then a ratio of the variances should equal 1. The F ratio is defined as: [pic]. The F ratio is a test statistic for determining the equality of variances, among other things. It is based on the F distribution (F sampling distribution). 2. Make the larger value the numerator, then...
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