In: Statistics and Probability
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.)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:reject the null hypothesisdo not reject the null hypothesis
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.
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.
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