Question

Akiko Hamaguchi is a manager at a small sushi restaurant in Phoenix, Arizona. Akiko is concerned that the weak economic environment has hampered foot traffic in her area, thus causing a dramatic decline in sales. In order to offset the decline in sales, she has pursued a strong advertising campaign. She believes advertising expenditures have a positive influence on sales. To support her claim, Akiko estimates the following linear regression model: Sales = β₀ + β₁Unemployment + β₂Advertising+ ε. A portion of the regression results is shown in the accompanying table.

ANOVA	df	SS	MS	F	Significance F
Regression	2	88.2574	44.1287	8.387	0.004
Residual	14	73.6638	5.2617
Total	16	161.9210

	Coefficients	Standard Error	t Stat	p-value
Intercept	33.1260	6.9910	4.7380	0.000
Unemployment	−0.6758	0.3459	−1.9540	0.071
Advertising	0.0287	0.0080	3.5880	0.003

a-1. Choose the appropriate hypotheses to test whether the explanatory variables jointly influence sales.

• H₀: β₁ = β₂ = 0; H_A: At least one β_j < 0

• H₀: β₁ = β₂ = 0; H_A: At least one β_j > 0

• H₀: β₁ = β₂ = 0; H_A: At least one β_j ≠ 0

a-2. Find the value of the appropriate test statistic. (Round your answer to 3 decimal places.)

Test statistic:_________

a-3. At the 5% significance level, do the explanatory variables jointly influence sales?

• Yes, since the F-test is significant.

• Yes, since all t-tests are significant.

• Both answers are correct.

b-1. Choose the hypotheses to test whether the unemployment rate is negatively related with sales.

• H₀: β₁ = 0; H_A: β₁ ≠ 0

• H₀: β₁ ≥ 0; H_A: β₁ < 0

• H₀: β₁ ≤ 0; H_A: β₁ > 0

b-2. Find the p-value.

• 0.025 p-value < 0.05
• 0.01  p-value < 0.025

• p-value < 0.01

• p-value  0.10
• 0.05 p-value < 0.10

b-3. At the 1% significance level, what is the conclusion to the test?

• Do not reject H₀; we can conclude that Unemployment is negatively related to Sales.

• Do not reject H₀; we cannot conclude that Unemployment is negatively related to Sales.

• Reject H₀; we can conclude that Unemployment is positively related to Sales.

• Reject H₀; we cannot conclude that Unemployment is positively related to Sales.

c-1. Choose the appropriate hypotheses to test whether advertising expenditures are positively related with sales.

• H₀: β₂ ≤ 0; H_A: β₂ > 0

• H₀: β₂ = 0; H_A: β₂ ≠ 0

• H₀: β₂ ≥ 0; H_A: β₂ < 0

c-2. Find the p-value.

• p-value < 0.01

• p-value  0.10
• 0.05  p-value < 0.10
• 0.025 p-value < 0.05
• 0.01 p-value < 0.025

c-3. At the 1% significance level, what is the conclusion to the test?

• Reject H₀  we can conclude that Advertising is positively related to Sales.
• Do not reject H₀ we can conclude that Advertising is positively related to Sales.
• Do not reject H₀  we cannot conclude that Advertising is positively related to Sales.
• Reject H₀ we cannot conclude that Advertising is positively related to Sales.

Answer 1

Result:

Akiko Hamaguchi is a manager at a small sushi restaurant in Phoenix, Arizona. Akiko is concerned that the weak economic environment has hampered foot traffic in her area, thus causing a dramatic decline in sales. In order to offset the decline in sales, she has pursued a strong advertising campaign. She believes advertising expenditures have a positive influence on sales. To support her claim, Akiko estimates the following linear regression model: Sales = β0 + β1Unemployment + β2Advertising + ε. A portion of the regression results is shown in the accompanying table.

ANOVA	df	SS	MS	F	Significance F
Regression	2	88.2574	44.1287	8.387	0.004
Residual	14	73.6638	5.2617
Total	16	161.9210

	Coefficients	Standard Error	t Stat	p-value
Intercept	33.1260	6.9910	4.7380	0.000
Unemployment	−0.6758	0.3459	−1.9540	0.071
Advertising	0.0287	0.0080	3.5880	0.003

a-1. Choose the appropriate hypotheses to test whether the explanatory variables jointly influence sales.

• H0: β1 = β2 = 0; HA: At least one βj ≠ 0

a-2. Find the value of the appropriate test statistic. (Round your answer to 3 decimal places.)

Test statistic: 8.387

a-3. At the 5% significance level, do the explanatory variables jointly influence sales?

• Yes, since the F-test is significant.

b-1. Choose the hypotheses to test whether the unemployment rate is negatively related with sales.

• H0: β1 ≥ 0; HA: β1 < 0

b-2. Find the p-value.

Since this is a one sided test, the required p value = 0.071/2 = 0.0355

• 0.025 < p-value < 0.05

b-3. At the 1% significance level, what is the conclusion to the test?

• Do not reject H0; we cannot conclude that Unemployment is negatively related to Sales.

c-1. Choose the appropriate hypotheses to test whether advertising expenditures are positively related with sales.

• H0: β2 ≤ 0; HA: β2 > 0

c-2. Find the p-value.

Since this is a one sided test, the required p value = 0.003/2 = 0.0015

• p-value < 0.01

c-3. At the 1% significance level, what is the conclusion to the test?

• Reject H0  we can conclude that Advertising is positively related to Sales.