Test Whether Drinking Coffee Increases Productivity

Question

A company wants to determine if drinking coffee increases productivity. A random sample of 50 employees measures productivity per day (# tasks completed) both before and after consuming coffee for 2 weeks. The mean of the differences (before - after) is 1.4 tasks with SD = 2.5 tasks.

Test whether drinking coffee increases productivity at a=.01 using the 4 step inference process.

Answer 1

Firstly, the question is written similarly to a two-sample problem. However, there are no givens (separate sample sizes and individual standard deviations or variances) to accomplish such a task. It is thus assumed that a one-sample inference here will suffice.

Step 1: Conceptualize the question.

We would like to know if the difference in productivity per day (marked by # tasks completed per day) is statistically different after consuming coffee versus before drinking coffee. Since we are only asked if there is a difference (no direction is specified) this is a bidirectional test. We will use a T-test to determine this (instead of a Z, T is appropriate if n< 400 and population variance is unknown)

Null Hypotheisis H0= Xbefore = XAfter 

Alternative Hypothesis Ha = Xbefore =/ XAfter

Decision rule: If T-score is outside the bounds of the t critical, then we have found a statistically significant difference

Step 2: Note your Givens

Xbar_before - Xbar_after= 1.4

SD = 2.5 

N = 50

A = .01 ( Two Sided alpha)

Degrees of freedom = n - 1 = 50 - 1 = 49

T Critical value(based on a=.01, using statistical appenix, df=49) = 2.680

Step 3: T-score formula Calculation

T = (Xbefore - Xafter) ÷ STD/Sqrt (n)

1. Convert the mean difference to a t-score
2. Compare the t-score to the t-critical values to determine if significance has been reached.

T = 1.4 ÷ 2.5/Sqrt (50)

T = 1.4 ÷ 0.03536 = 39.5927

Step 4: Compare/Interpret/Conclude

(-Tcrit < Ttest < +Tcrit) = (-2.680 < 39.5927 < +2.680)

Is 39.59 between -2.680 and +2.680.

No. 

So we have evidence to reject the null hypothesis that the mean number of tasks completed before drinking coffee is the same as the mean number of tasks completed after drinking coffee. Specifically, we have evidence to suggest that the mean number of tasks before drinking coffee is greater than after drinking coffee. 

Why?

Because before - after = 1.4 implies that the amount of tasks completed before drinking coffee must be greater than the number of tasks completed after, and we found that difference to be significant via T-test.

Update: According to these results, we do not have evidence to suggest that coffee increases productivity; in fact, we found significant results that it may decrease productivity, as tasks completed before were significantly higher than tasks completed after coffee consumption.

Conclusion: Evidence shows that coffee consumption does not increase productivity.