Question

We want to investigate the diameter of steel rods that are manufactured on two different sites. We pick two different random samples of sizes n1 = n2 = 15. The sample means are X1 = 6.2, X2 = 7.8, respectively. The sample variances are s 2 1 = 4 and s 2 2 = 6.25. Assume that both sites produce rods of diameter that is normally distributed with the same standard deviation σ1 = σ2. Answer the following questions.

(a) Use α = 0.05 to accept or reject the hypothesis that both sites produce the same diameter rods.

(b) Assume that the two sites are unsure about whether their standard deviations are actually the same. Using the data in the exercise and assuming α = 0.05, should you accept or reject the hypothesis that both sites have the same standard deviation?

Answer 1

Theory:

For,

Equal sample sizes, equal variance [edit] Given two groups (1, 2), this test is only applicable when: • the two sample sizes (that is, the number n of participants of each group) are equal; . it can be assumed that the two distributions have the same variance; Violations of these assumptions are discussed below. The t statistic to test whether the means are different can be calculated as follows: t-X1 - X2 where sp= V2 Here Sy is the pooled standard deviation for n=nı = n2 and sy, and sy are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means. For significance testing, the degrees of freedom for this test is 2n – 2 where n is the number of participants in each group.

So

S_p= 2.26

t= (6.2-7.8)/(2.26*0.365)

=-1.686

P Value Results

t=-1.686   DF=28   

   The two-tailed P value equals 0.1029

So We reject the Null Hypothesis at alfa=0.05.

Theory:

For,

Equal or unequal sample sizes, unequal variances [edit] Main article: Welch's t-test This test, also known as Welch's t-test, is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The t statistic to test whether the population means are different is calculated as: X - X2 t= SA where Here sí? is the unbiased estimator of the variance of each of the two samples with n; = number of participants in group i (1 or 2). In this Sa is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's t-distribution with the degrees of freedom calculated using 22 d. f. = + (s/n)? 12-1 ni-1 This is known as the Welch-Satterthwaite equation. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see Behrens-Fisher problem).

So

SΔ= 1.9159

t= (6.2-7.8)/ 1.9159

=-0.835117

P Value Results

t=-0.835117   DF=23.82   

   The two-tailed P value equals 0.4119

So We reject the Null Hypothesis at alfa=0.05