# A Step-by-Step Introduction to Statistics for Business

# Chapter 11: Chi-squared (χ2) tests of fit

Answers for **Data Skill Challenges** for every chapter in the book can be found to check your performance and widen your understanding.

**RQ**: Are any of the four gums chosen more often by focus group participants?

*H*_{0}: There is no difference between observed and expected frequencies of gum preference.

*H*_{1}: There is a difference between observed and expected frequencies of gum preference.

*α* = .05

*χ*^{2}_{crit}(3) = +7.815

*χ*^{2}(3) = +10.17, *p* = .02 or *χ*^{2}(3) = +10.17, *p *<.05

φ* _{C}* = 0.33

**Conclusion**: Reject the null and accept the alternative. The chi-squared is statistically significant. There is a difference between observed and expected frequencies of gum preference. Cramérs V is 0.33. Gum D is picked most often, while the other three gums are roughly the same.

2) **RQ: **Are any of the logos chosen more often by the employees?

*H*_{0}: There is no difference between observed and expected frequencies of logo preference.

*H*_{1}: There is a difference between observed and expected frequencies of logo preference.

*α* = .05

*χ*^{2}(2)_{crit} = +5.991

*χ*^{2}(2) = 3.75, *p* = .154 or *χ*^{2}(2) = 3.75, *p* > .05

**Conclusion: **Retain the null. The chi-squared is not statistically significant. There is insufficient evidence that there is a difference between observed and expected frequencies of logo preference.

3) **RQ: **Are there more accidents on night shift for some plants?

*H*_{0}: Accidents by plant are independent of shift.

*H*_{1}: Accidents by plant are not independent of shift.

*α* = .05

*χ*^{2}_{crit}(11) = +19.675

*χ*^{2}(11) = +21.961, *p* = .03 or c^{2}(11) = +21.961, *p *<.05

φ* _{C}* = 0.46

**Conclusion: **Reject the null and accept the alternative. The chi-squared is statistically significant. Accidents by plant are not independent of shift. Cramérs V is 0.46. Plants 9, 10 and 12 have noticeably higher accident rates on the night shift than wed expect by shift or plant alone.

**4) RQ: **Are there more sales for food for some colour carts?

*H*_{0}: Sales by food type are independent of cart colour.

*H*_{1}: Sales by food type are not independent of cart colour.

*α* = .05

*χ*^{2}(2)_{crit} = 5.991

*χ*^{2}(2) = 2.68, *p* = .26 or *χ*^{2}(2) = 2.68, *p* > .05

**Conclusion: **Retain the null. The chi-squared is not statistically significant. There is insufficient evidence to conclude that sales by food type are not independent of cart colour.