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

# Chapter 12: Correlation and Regression

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

1) **RQ: **Is there a linear relationship in the ratings of Dish 2 and Dish 7?

*H*_{0}: ρ = 0

*H*_{1}: ρ ≠ 0

*α* = .05

*t*_{crit}(18) = ±2.10

*r*(18) = 0.33, *p *>.05

**Conclusion**: Retain the null. The correlation is not statistically significant. There is not a linear relationship between client costs and percentage increase in profits.

- Is there a linear relationship between July and August sales?

*H*_{0}: ρ = 0

*H*_{1}: ρ ≠ 0

*α* = .05

*t*(50)_{crit} = ±2.01

*r*(52) = 0.87, *p* < .05

*r ^{2} = *0.76

*y*' = 0.93*x* + 0.71

*y*' = 0.93(14) + 0.71 = 13.66 = 14

**Conclusion**: Reject the null and accept the alternative. The correlation is statistically significant. There is a linear relationship between July and August. 76% of the variance in August sales was explained by the variance in July sales. For every car sold in July, we’d expect 0.93 additional cars sold in August. If a salesperson sold 14 cars in July, we’d expect that person to sell 13.66 (rounded to 14) cars in August.

3) **RQ: **Is there a linear relationship between the temperature and number of customers?

*H*_{0}: ρ = 0

*H*_{1}: ρ ≠ 0

*α* = .05

*t*_{crit}(12) = +1.782

*r*(12) = 0.90, *p *<.05

*r*^{2} = 0.81

*y*' = 2.222*x* − 26.51

**Conclusion**: Reject the null and accept the alternative. The correlation is statistically significant. There is a linear relationship between the temperature and number of customers. 81% of the variance in the number of customers was explained by the temperature. For every degree increase in temperature, we’d expect 2.22 additional customers. If the temperature is 38 degrees, he should expect 57.93 (58) customers.

4) **RQ: **Is there a linear relationship in the number of hours worked and the number of cells entered?

*H*_{0}: ρ = 0

*H*_{1}: ρ ≠ 0

*α* = .05

*t*(14)_{crit} = ±2.145

*r*(14) = 0.60, *p* > .05

**Conclusion**: Retain the null. The correlation is not statistically significant. There is not a linear relationship between the number of hours worked and the number of cells entered.