In: Statistics and Probability
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 From 99 of its restaurants, Noodles & Company managers collected data on per-person sales and the percent of sales due to "potstickers" (a popular food item). Both numerical variables failed tests for normality, so they tried a chi-square test. Each variable was converted into ordinal categories (low, medium, high) using cutoff points that produced roughly equal group sizes. At α = .02, is per-person spending independent of percent of sales from potstickers?  | 
| Potsticker % of Sales | ||||
|   Per Person Spending  | 
Low | Medium | High | Row Total | 
| Low | 18 | 7 | 8 | 33 | 
| Medium | 10 | 11 | 5 | 26 | 
| High | 12 | 9 | 19 | 40 | 
| Col Total | 40 | 27 | 32 | 99 | 
| (a) | 
 The hypothesis for the given issue is H0: Percentage of Sales and Per-Person Spending are independent.  | 
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| (b) | 
 Calculate the chi-square test statistic, degrees of freedom, and the p-value. (Round your test statistic value to 2 decimal places and p-value to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)  | 
| Test statistic | |
| d.f. | |
| p-value | |
| (c) | We reject the null and find dependence. | ||||
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Solution:
(a) The hypothesis for the given issue is H0: Percentage of Sales and Per-Person Spending are independent.
Answer: Yes
(b) Calculate the chi-square test statistic, degrees of freedom, and the p-value.
Answer:
| Test statistic | ![]()  | 
| df | ![]()  | 
| p-value | ![]()  | 
(c)
We reject the null and find dependence.
Answer: No
Because the p-value is greater than the significance level 0.02
Explanation:

The observed data is given below:
| Per Person Spending | Low | Medium | High | Row Total | 
| Low | 18 | 7 | 8 | 33 | 
| Medium | 10 | 11 | 5 | 26 | 
| High | 12 | 9 | 19 | 40 | 
| Col Total | 40 | 27 | 32 | 99 | 
The expected data is calculated using the below formula:

| Per Person Spending | Low | Medium | High | Row Total | 
| Low | 13.33333 | 9 | 10.66667 | 33 | 
| Medium | 10.50505 | 7.090909 | 8.40404 | 26 | 
| High | 16.16162 | 10.90909 | 12.92929 | 40 | 
| Col Total | 40 | 27 | 32 | 99 | 
The chi-square value for each cell is:
| Per Person Spending | Low | Medium | High | Row Total | 
| Low | 1.633333 | 0.444444 | 0.666667 | 2.744444 | 
| Medium | 0.024281 | 2.155012 | 1.3788 | 3.558093 | 
| High | 1.071616 | 0.334091 | 2.850387 | 4.256094 | 
| Col Total | 2.729231 | 2.933547 | 4.895853 | 
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