Question

A sample of 7 department store sizes (in thousands of square feet) and revenues ( in million of dollars) are found to have a linear coefficient of 0.445. Find the critical valuesfor the linear correlation coefficient, assuming a 0.05 significance level. Is there sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores?

Answer 1

Solution

Back-up Theory

To test for significance of correlation coefficient, i.e.,

Null: H₀: ρ = 0 Vs Alternative H₁: ρ ≠ 0 ....................................................................................................... (1)

Test Statistic:

t = r√{(n - 2)/(1 – r²)} .................................................................................................................................. (2)

Distribution, Significance Level, α, Critical Value, p-value

t ~ t_{n – 2}

So, Critical Value = upper (α/2) % point of t_{n – 2}....................................................................................................... (3)

p-value = P(t_{n – 2} > tcal) ....................................................................................................... (3a)

Decision

H₀ is rejected/accdepted if | t_cal | > < t_crit ....................................................................................................... (4)

or equivalently, if p-value < > α ....................................................................................................... (4a)

To work out the solution

Given n = 7, r = 0.0.445 and significance level, α = 0.05

Part (a)

Vide (3), critical value

= upper 2.5% of t₅

= 2.571 [from standard t-Table] Answer 1

Part (b)

Vide (2),

Test Statistic: t = 0.445√{5/(1 – 0.198025)}

= 1.1111

Vide (4),

H₀ is accdepted since | t_cal | < t_crit

Hence we infer that there is not sufficient evidence to conclude that the linear correlation between size and revenue is significant. Answer 2

DONE