Question

The management of Discount Furniture, a chain of discount furniture stores in the Northeast, designed an incentive plan for salespeople. To evaluate this innovative plan, 14 salespeople were selected at random, and their weekly incomes before and after the plan were recorded. (use the six steps of hypothesis testing)

Salesperson	Before	After
1	580	615
2	562	636
3	618	633
4	611	627
5	600	687
6	603	698
7	563	665
8	584	599
9	564	678
10	600	662
11	606	718
12	563	716

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d> 0

Alternative hypothesis: u_d < 0

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = sqrt [ (\sum (d_i - d)² / (n - 1) ]

s = 45.3919

SE = s / sqrt(n)

S.E = 13.104

DF = n - 1 = 12 -1

D.F = 11

t = [ (x₁ - x₂) - D ] / SE

t = - 5.59

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 11 degrees of freedom is less than - 5.59.

Thus, the P-value = less than 0.0001

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.