Question

"To Breakfast or Not to Breakfast?" by Richard Ayore

In the American society, birthdays are one of those days that everyone looks forward to. People of different ages and peer groups gather to mark the

18^th, 20^th,   ,

birthdays. During this time, one looks back to see what he or she has achieved for the past year and also focuses ahead for more to come.

If, by any chance, I am invited to one of these parties, my experience is always different. Instead of dancing around with my friends while the music is booming, I get carried away by memories of my family back home in Kenya. I remember the good times I had with my brothers and sister while we did our daily routine.

Every morning, I remember we went to the shamba (garden) to weed our crops. I remember one day arguing with my brother as to why he always remained behind just to join us an hour later. In his defense, he said that he preferred waiting for breakfast before he came to weed. He said, "This is why I always work more hours than you guys!"

And so, to prove him wrong or right, we decided to give it a try. One day we went to work as usual without breakfast, and recorded the time we could work before getting tired and stopping. On the next day, we all ate breakfast before going to work. We recorded how long we worked again before getting tired and stopping. Of interest was our mean increase in work time. Though not sure, my brother insisted that it was more than two hours. Using the data in the table below, solve our problem. (Use

α = 0.05)

• State the distribution to use for the test. (Enter your answer in the form z or t_df where df is the degrees of freedom.)

• Part (e)

  What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)

• Part (f)

  What is the p-value?

Work hours with breakfast	Work hours without breakfast
8	6
6	5
10	6
5	4
9	7
8	7
10	7
7	5
6	6
9	5

Answer 1

Following is the raw dataset:

no breakfast	breakfast
6	8
5	6
6	10
4	5
7	9
7	8
7	10
5	7
6	6
5	9

Let, Sample of work hours with breakfast be denoted by A

and, Sample of work hours without breakfast be denoted by B

given, Sample size, n = 10

and, level of significance, α = 0.05

Let , Di be the difference between the corresponding samples, given by:

Di = Bi - Ai , for ith observation.

So, we get the following table:

breakfast	no breakfast	difference
8	6	-2
6	5	-1
10	6	-4
5	4	-1
9	7	-2
8	7	-1
10	7	-3
7	5	-2
6	6	0
9	5	-4

now, total difference, D = ΣDi = -20

State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.) ?

Here, since the population standard deviation σ is unknown, we will be using the student's t-distribution for testing our hypothesis.

Also, we will perform a paired t-test for difference in mean as the two samples are taken from the same experiment subjects  where, degree of freedom is ginen by:

DOF = n - 1 = 20 - 1 = 9

What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)

also,

Null Hypothesis, Ho : μD < 2

Alternate Hypothesis, HA : μD > 2

where, μD = μB - μA

Here, the test statistic we use is tSTAT, which is calculated using the formula :

tSTAT =(( Dbar - μD) / ( sD / (n)1/2 )) = 1.334

where,

Dbar =(D / n) = -20 / 10 = -2 ,

μD = 2 (assumed mean difference)

n = 10

and,

sD =((D - Dbar) / (n-1))1/2

Calculating the tSTAT:

tSTAT =(( Dbar - μD) / ( sD / (n-1)1/2 )) = (( -2 - 2 ) / ( 1.334 / 3.162) ) = -9.481

Here, since |tSTAT | > 1.833 at tdof = 9 , so we reject the Null hypothesis , Ho, and state that

"the working hours with a breakfast increases by more than two hours"

What is the p-value?

From p-value approach, using tSTAT table, we found out that :

for the given Degrees of Freedom

The p-value is < .00001.