Question

In: Statistics and Probability

A major food manufacturer is concerned that the sales for its skinny french fries have been...

A major food manufacturer is concerned that the sales for its skinny french fries have been decreasing. As a part of a feasibility study, the company conducts research into the types of fries sold across the country to determine if the type of fries sold is independent of the area of the country. The results of the study are shown in the table below. Conduct a test of independence. (Use a significance level of 0.05.)

Type of Fries Northeast South Central West
skinny fries 69 51 20 24
curly fries 99 60 15 30
steak fries 20 40 10 11
  • Part (a)

    State the null hypothesis.

    The types of fries sold are dependent on the location.The types of fries sold are independent of the location.    

  • Part (b)

    State the alternative hypothesis.

    The types of fries sold are dependent on the location.The types of fries sold are independent of the location.    

  • Part (c)

    What are the degrees of freedom? (Enter an exact number as an integer, fraction, or decimal.)
    (No Response)

  • Part (d)

    State the distribution to use for the test.

    t6

    χ26

         

    t2

    χ22

  • Part (e)

    What is the test statistic? (Round your answer to two decimal places.)
    (No Response)

  • Part (f)

    What is the p-value? (Round your answer to four decimal places.)
    (No Response)

    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 value of the test statistic will be equal to or greater than the calculated value.If H0 is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.    If H0 is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.If H0 is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.

  • 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) and write the appropriate conclusion.(i) Alpha:
    α = (No Response)

    (ii) Decision:

    reject the null hypothesisdo not reject the null hypothesis    


    (iii) Reason for decision:

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


    (iv) Conclusion:

    There is sufficient evidence that types of fries and location are dependent.There is not sufficient evidence that types of fries and location are dependent.   

Solutions

Expert Solution

Northeast   South   Central   West   Total  
skinny fries Observed   69 51 20 24 164
Expected   68.67 55.15 16.44 23.74 164.00
O - E   0.33 -4.15 3.56 0.26 0.00
(O - E)² / E   0.00 0.31 0.77 0.00 1.09
curly fries Observed   99 60 15 30 204
Expected   85.42 68.61 20.45 29.53 204.00
O - E   13.58 -8.61 -5.45 0.47 0.00
(O - E)² / E   2.16 1.08 1.45 0.01 4.70
steak fries Observed   20 40 10 11 81
Expected   33.92 27.24 8.12 11.73 81.00
O - E   -13.92 12.76 1.88 -0.73 0.00
(O - E)² / E   5.71 5.98 0.44 0.04 12.17
Total Observed   188 151 45 65 449
Expected   188.00 151.00 45.00 65.00 449.00
O - E   0.00 0.00 0.00 0.00 0.00
(O - E)² / E   7.87 7.37 2.66 0.06 17.95

Part (a)

State the null hypothesis.

The types of fries sold are independent of the location.    

Part (b)

State the alternative hypothesis.

The types of fries sold are dependent on the location.

Part (c)

What are the degrees of freedom? (Enter an exact number as an integer, fraction, or decimal.)

6

Part (d)

State the distribution to use for the test.

χ26

  

Part (e)

What is the test statistic? (Round your answer to two decimal places.)
17.95

Part (f)

What is the p-value? (Round your answer to four decimal places.)
0.0063



Explain what the p-value means for this problem.

.If H0 is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.


Part (h)

Indicate the correct decision ("reject" and write the appropriate conclusion.(i) Alpha:
α = 0.05

(ii) Decision:

reject the null hypothesis


(iii) Reason for decision:

.Since α > p-value, we reject the null hypothesis.


(iv) Conclusion:

There is sufficient evidence that types of fries and location are dependent


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