The management of Discount Furniture, a chain of discount...

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, 12 salespeople were selected at random, and their weekly incomes before and after the plan were recorded. Was there a significant increase in the typical salesperson's weekly income due to the innovative incentive plan? Use the .05 significance level. Estimate the $p$ -value, and interpret it.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that involves making an initial assumption, called the null hypothesis, and then using sample data to decide whether there is sufficient evidence to reject this assumption. It is a systematic way to determine whether an observed effect, such as a change in performance, is likely to be genuine or a result of random variation in the data.

Paired T-Test

A paired t-test is used to compare the means of two related groups. In cases where the same subjects are measured twice (for example, before and after an intervention), the test analyzes the differences in paired observations rather than comparing two independent groups. This helps control for individual variability, making the test more sensitive to detecting true changes.

p-value

The p-value is the probability of obtaining results at least as extreme as those observed, assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis—a small p-value suggests that the observed data are unlikely under the null hypothesis, thus providing support for the alternate hypothesis.

Significance Level

The significance level, commonly set at 0.05, is the threshold used in hypothesis testing to judge whether the p-value indicates a statistically significant result. If the p-value is less than this predetermined level, the evidence is considered strong enough to reject the null hypothesis, suggesting that the observed effect is statistically significant.

Transcript

00:02 So we have 12 people who we're going to be concerned with, and i'm going to let the difference stand for the after salary minus the before.

00:11 We want to see if those incentives have increased their sales.

00:17 So we will be assuming that that mean difference is equal to zero, and alternately that after they have had that incentive, that they sell, they make more per week.

00:28 And our sample, i have my values, if i subtracted correctly, are 20, negative 5, 54, 0, 98, 6, and 0, 5, and 0, 19, and 114.

00:58 So now we do have, they appear that there is an increase, but we do see that we have some zeros and we have some significantly big numbers here.

01:07 So in any case, when i get those values listed down, i find out that the, let me quick put this in and find what the mean is for our situation.

01:21 And then we will do our test.

01:23 And we will be using a t test since we don't know the population standard ev.

01:29 And we have these small sample sizes.

01:32 So we have our x bar for those differences comes out to be 25 .916 repeating and our sample standard deviation for those is also going to be relatively large because we do have quite a bit of variability going on here.

01:50 We have 40 .79 it would round to one.

01:53 And so if we're using our critical value, we would be looking at that t distribution and finding with 5 % having all 5 % in that upper tail, and we'd want to find that value that has 11 degrees of freedom and 5 % in the upper tail.

02:11 And 11 degrees of freedom with 5 % is a t value of 1 .76.

02:17 So anything that is higher than that would cause us to reject our null.

02:23 And we would fail to reject if it's on the other side.

02:26 And so let's calculate that test that test statistic with 11 degrees of freedom comes out to be our mean minus the mean we're assuming and then that sample standard deviation, which again is pretty big divided by the square root of 12.

02:45 And that test statistic is 2 .209.

02:49 And we can see that that does lie up here...

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