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. $$ \begin{array}{|lrr|} \hline \text { Salesperson } & \text { Before } & \text { After } \\ \hline \text { Sid Mahone } & \$ 320 & \$ 340 \\ \text { Carol Quick } & 290 & 285 \\ \text { Tom Jackson } & 421 & 475 \\ \text { Andy Jones } & 510 & 510 \\ \text { Jean Sloan } & 210 & 210 \\ \text { Jack Walker } & 402 & 500 \\ \text { Peg Mancuso } & 625 & 631 \\ \text { Anita Loma } & 560 & 560 \\ \text { John Cuso } & 360 & 365 \\ \text { Carl Utz } & 431 & 431 \\ \text { A. S. Kushner } & 506 & 525 \\ \text { Fern Lawton } & 505 & 619 \\ \hline \end{array} $$ 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.

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.
$$
\begin{array}{|lrr|}
\hline \text { Salesperson } & \text { Before } & \text { After } \\
\hline \text { Sid Mahone } & \$ 320 & \$ 340 \\
\text { Carol Quick } & 290 & 285 \\
\text { Tom Jackson } & 421 & 475 \\
\text { Andy Jones } & 510 & 510 \\
\text { Jean Sloan } & 210 & 210 \\
\text { Jack Walker } & 402 & 500 \\
\text { Peg Mancuso } & 625 & 631 \\
\text { Anita Loma } & 560 & 560 \\
\text { John Cuso } & 360 & 365 \\
\text { Carl Utz } & 431 & 431 \\
\text { A. S. Kushner } & 506 & 525 \\
\text { Fern Lawton } & 505 & 619 \\
\hline
\end{array}
$$
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

Paired Sample Design

A paired sample design involves collecting two sets of related observations – one from each condition or time point – on the same subjects. This method is used to control for individual differences between subjects, allowing for a more precise estimation of the treatment or intervention effect by analyzing the differences within each paired observation.

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide whether there is enough evidence in a sample to infer that a particular condition is true for the entire population. In this context, the null hypothesis typically states that there is no change in weekly income due to the incentive plan, while the alternative hypothesis suggests that the income has significantly increased.

Paired t-Test

The paired t-test is a statistical procedure applicable when comparing the means of two related groups, such as measurements taken before and after an intervention. It involves calculating the differences between each pair of observations, assuming these differences follow a normal distribution, and then testing whether the mean difference is significantly different from zero.

Significance Level

The significance level, often denoted as ? (alpha), is a threshold used to decide whether the observed data are unlikely under the null hypothesis. A common significance level is 0.05, meaning there is a 5% risk of rejecting the null hypothesis when it is actually true.

p-Value Interpretation

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the value observed, assuming that the null hypothesis is true. A small p-value (typically less than the significance level) indicates strong evidence against the null hypothesis, suggesting that the effect of the intervention, such as an increase in income, is statistically significant.

Transcript

00:01 So we have data before and after, and i'm having my difference be the after the implement that incentive plan minus the before.

00:09 That's good to state which way you're subtracting.

00:13 And so we have a sample size of 12, and we want to assume that that mean of the d is going to be either equal to zero or you may want to have it as less than or equal to zero, depending on the way your instructor has.

00:28 And then alternately, we think that actually that incentive plan has made that be after b higher than before.

00:35 So let's find what our data is for the mean and standard deviation.

00:42 And again, i have all of those 12 subtracted, and i'll tell you what the mean is for that particular setting.

00:49 And almost set.

00:54 Just had to quick type in a couple things.

00:57 All right.

00:57 So i have that the mean of those 12 numbers comes out to be 25.

01:01 Point nine one and it's actually six repeating and then the standard deviation is 40 .791 and so we would be assuming that that sampling distribution of 12 means that that would center at zero and we're getting this mean up 20 something 25 point something 25 .9 i'll just say two and this area here is we know our p value.

01:35 And so let's find determine what that p value is and find out how likely it is for that type of thing to happen in this distribution.

01:43 So we need to determine what the test statistic is, which is a t value with 11 degrees of freedom.

01:49 And we're going to take what we got minus the mean we're assuming and that standard deviation, which is relatively big divided by the square root of n, which is that 12.

02:02 And that test statistic comes out to be 2 .2 .2 .7...

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