In: Statistics and Probability
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 |