Chapter 10 indepedent t test
The t statistic for the independent-measures t test is
t = (M1 − M2) − (μ1 − μ2)s(M1 − M2)
Since the null hypothesis is that the two population means are the same, μ₁ - μ₂ = 0. It can be calculated as:
t = (M1 − M2)s(M1 − M2)(M1 − M2)s(M1 − M2)
Which of the following null hypotheses is appropriate for an independent-measures t test?
H₀: μ₁ - μ₂ = 0 ypically, the null hypothesis for the independent-measures t test states that there is no difference between the means of two different populations. Specifically, the null hypothesis refers to two different means that are subscripted to denote the different population means, such as μ₁ and μ₂ or μcontrolμcontrol and μtreatmentμtreatment. A null hypothesis such as H₀: μ₁ - μ₂ = 0 is equivalent to the null hypothesis H₀: μ₁ = μ₂ in that both indicate that the means of the two populations are the same. Often, the version that shows the difference is used, because the t statistic actually uses this hypothesized difference. Although the hypothesized difference is almost always zero (making H₀: μ₁ = μ₂ just as appropriate), the independent-measures t test can be used to test, for example, whether this difference is a particular nonzero value (such as 2). A hypothesis such as H₀: μ ≥ 0 or H₀: μ < 4 refers to only one mean and would be appropriate for the single-sample t test. Also, a null hypothesis always must have an equal sign, therefore, hypotheses such as H₀: μ < 4 are appropriate alternative hypotheses for a single-sample t test. The hypothesis H₀: μ₁ - μ₂ ≠ 0 would be an appropriate alternative hypothesis for the independent-measures t test, as it indicates that there is an effect, specifically that the means are different.
In the independent-measures t test the difference between the means of the two groups is divided by the estimated standard error expected if the groups came from the same population.
The estimated error is calculated from the variances of the two groups. If this ratio is more than is expected by chance for groups their size, the groups are considered to be significantly different from one another.
independent samples
The experiment that compares the weight gain of premature babies to the weight gain of full-term babies has two independent and unrelated samples of babies. The experiment that compares depression scores of a single sample of girls before and after they attend a self-image camp uses the same sample of people to compare two repeated measures. The experiment that compares depression scores of girls who attend a self-image camp to the known Hamilton score for teenage girls uses one sample and compares it to a pre-specified mean.
