Question

Given the following information, calculate the pooled sample variance. Round your answer to 2 decimal places. S1 3 S2 = 5 n1 37 n2 30

Answer 1

1. The pooled standard deviation is given by:

2 2 (n1-1)s1(n2- 1)s5 Sp ni n2- 2

(37 1)32 (30 1)52 37 30 2

= 4.0173

2. The results are computed in the table given below

Subjects	Weight before	Weight after	Weight change (d)
1	285	253	-32
2	265	230	-35
3	302	285	-17
4	293	298	5
5	269	255	-14
6	288	271	-17
			$\sum$d = -110

Average weight change =

n

=

-110 6

= -18.33

3.

The provided sample means are shown below:

X112

X213.5

Also, the provided sample standard deviations are:

s₁​=2.5 s₂​=2.3

and the sample sizes are n₁​=18 and n_2​ =18

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:μ₁​ = μ₂​ ( no difference)

Ha: μ_1​ ≠ μ_2​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

Xi - X2 t= (n1-1)s+(n 1)s ( n1+n2-2 12 13.5 = -1.873 (18-1)2.52 (18 1)2.3 (+18 18+18-2

At significance level α=0.05, and the degrees of freedom are df = 34,  the critical value for this two-tailed test is

t_c​=2.032.

Since it is observed that |t| = 1.873 ≤ t_c​=2.032, it is then concluded that the null hypothesis is not rejected.

Hence the sale of two varieties of truffles differ significantly during the holiday season.