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)

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

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

• Part (a)

  State the null hypothesis.

  H₀: μ_d > 0

  H₀: μ_d ≠ 0

      

  H₀: μ_d < 0

  H₀: μ_d ≥ 0

  H₀: μ_d = 0

• Part (b)

  State the alternative hypothesis.

  H_a: μ_d < 0

  H_a: μ_d ≤ 0

      

  H_a: μ_d > 0

  H_a: μ_d ≥ 0

  H_a: μ_d ≠ 0

• Part (c)

  In words, state what your random variable

  X_d

  represents.

  X_d

  represents the average work times of the 10 days.

  X_d

  represents the average difference in work times on days when eating breakfast and on days when not eating breakfast.    

  X_d

  represents the total difference in work times on days when eating breakfast and on days when not eating breakfast.

  X_d

  represents the difference in the average work times on days when eating breakfast and on days when not eating breakfast.

• Part (d)

  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.)
  ---Select--- z t =

• Part (f)

  What is the p-value?

  p-value < 0.0100.010 < p-value < 0.050    0.050 < p-value < 0.100p-value > 0.100

  
  
  Explain what the p-value means for this problem.If

  H₀

  is false, then there is a chance equal to the p-value that the sample average difference between work times on days when eating breakfast and on days when not eating breakfast is less than 2.1.If

  H₀

  is true, then there is a chance equal to the p-value that the sample average difference between work times on days when eating breakfast and on days when not eating breakfast is less than 2.1.    If

  H₀

  is true, then there is a chance equal to the p-value that the sample average difference between work times on days when eating breakfast and on days when not eating breakfast is at least 2.1.If

  H₀

  is false, then there is a chance equal to the p-value that the sample average difference between work times on days when eating breakfast and on days when not eating breakfast is at least 2.1.

• Part (g)

  Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.

• Part (h)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
  α =
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since p-value < α, we reject the null hypothesis.Since p-value > α, we do not reject the null hypothesis.    Since p-value > α, we reject the null hypothesis.Since p-value < α, we do not reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the mean difference in work times on days when eating breakfast and on days when not eating breakfast has increased.There is not sufficient evidence to conclude that the mean difference in work times on days when eating breakfast and on days when not eating breakfast has increased.    

• Part (i)

  Explain how you determined which distribution to use.

  The t-distribution will be used because the samples are dependent.The standard normal distribution will be used because the samples involve the difference in proportions.    The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known.

Answer 1