Q. 91 A major food manufacturer is con... [FREE SOLUTION]

91影视

Introductory Statistics

Barbara Illowsky, Susan Dean

$Math Studyset 91影视 Explanations$ Math

OER 2018 Edition 路 902 pages

ISBN: 9781938168208

Chapter 11: Q. 91 (page 664)

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 Table 11.48. Conduct a test of independence.

Short Answer

Expert verified

The type of fries sold is dependent of the area of the country.

Step by step solution

Step 1. Given information

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 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 Table 11.48.

Step 2. Calculation

Type of Fries	Northeast	South	Central	West	Row Total
Skinny fries	70	50	20	25	165
Curly fries	100	60	15	30	205
Steak fries	20	40	10	10	260
Column Total	370	150	45	65	630

The two factors are the type of fries and the area of the country.

$H_{0}$: The type of fries sold is independent of the area of the country

$H_{1}$: The type of fries sold is dependent of the area of the country

The expected results and the value of the test statistic are shown in the table.

Observed	Expected	(Obs-Exp)	(Obs-Exp)^2	(Obs-Exp)^2/Exp
70	96.90	-26.90	723.87	7.47
100	120.40	-20.40	416.03	3.46
200	152.70	47.30	2237.44	14.65
50	39.29	10.71	114.80	2.92
60	48.81	11.19	125.23	2.57
40	61.90	-21.90	479.82	7.75
20	11.79	8.21	67.47	5.73
15	14.64	0.36	0.13	0.01
10	26.83	-16.83	283.09	10.55
25	17.02	7.98	63.62	3.74
30	21.15	8.85	78.31	3.70
10	26.83	-16.83	283.09	10.55
				73.10

Therefore, value of the test statistic: $X^{2}=73.10$

Distribution for the test: $X^{2}_{6}$

$df=(4columns -1)(3rows-1)=3\times 2=6$

Graph:

Probability statement: p-value$=P(x^{2}>73.10)=3.1379\times 10^{-11}$

Compare $\alpha$ and the p-value: Since no $\alpha$ is given, assume $\alpha=0.05$. p-value$=3.1379\times 10^{-11}$.

Therefore, $\alpha >p-$value.

Decision: Since $\alpha >p-$value, reject $H_{0}$. This means that the factors are not independent.

Conclusion: At a $5%$ level of significance, from the data, there is sufficient evidence to conclude that the type of fries sold and the area of the country are dependent on one another.

The type of fries sold is dependent of the area of the country.