Mathematics

value:
25.00 points
The following ANOVA table was obtained when estimating a multiple linear regression model. Use Table 4.
  ANOVA df SS MS F Significance F
  Regression 2 22,894.43 11,447.22 ? 0.0207
  Residual 17 39,588.56 2,328.74
  Total 19 62,482.99

a-1. How many explanatory variables were specified in the model?
  Number of explanatory variables [removed]
a-2. How many observations were used?
  Number of observations [removed]
  b. Choose the appropriate hypotheses to determine whether the explanatory variables are jointly significant.
[removed] H0β1 = β2 = 0; HA: At least one β j ≠ 0
[removed] H0β1 = β2 = 0; HA: At least one β j > 0
[removed] H0β1 = β2 = 0; HA: At least one β j < 0
  c. Compute the value of the test statistic. (Round your answer to 2 decimal places.)
  Test statistic [removed]
d-1. Find the p-value. (Round your answer to 4 decimal places.)
  p-value [removed]
d-2. At the 5% significant level, what is the conclusion to the test?
[removed] Reject H0Picture the explanatory variables are jointly significant in explaining y.
[removed] Reject H0Picture the explanatory variables are not jointly significant in explaining y.
[removed] Do not reject H0Picture the explanatory variables are jointly significant in explaining y.
[removed] Do not reject H0Picture the explanatory variables are not jointly significant in explaining y.

 

 

 

2.

value:
25.00 points
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 assumes the linear regression model as Sales = β0 + β1 Advertising + βUnemployment + ε. A portion of the regression results is shown in the accompanying table. Use Table 2 and Table 4.

 

  ANOVA df SS MS F Significance F
  Regression 2 88.2574 44.1287 8.387 0.0040
  Residual 14 73.6638 5.2617
  Total 16 161.9212

 

Coefficients Standard Error t Stat p-value Lower 95% Upper 95%
  Intercept 33.1260 6.9910 4.7384 0.0003 18.1300 48.12
  Advertising 0.0287 0.0080 3.5875 0.0029 0.0100 0.05
  Unemployment −0.6758 0.3459 −1.9537 0.0710 −1.4200 0.0700

 

a-1. Choose the appropriate hypotheses to test whether the explanatory variables jointly influence sales.
[removed] H0β1 = β2 = 0; HA: At least one β j < 0
[removed] H0β1 = β2 = 0; HA: At least one β j > 0
[removed] H0β1 = β2 = 0; HA: 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 [removed]

 

a-3. At the 5% significance level, do the explanatory variables jointly influence sales?
[removed] Yes, since the F-test is significant.
[removed] Yes, since all t-tests are significant.
[removed] Both answers are correct.

 

b-1. Choose the hypotheses to test whether the unemployment rate is negatively related with sales.
[removed] H0β2 = 0; HAβ2 ≠ 0
[removed] H0β2 ≤ 0; HAβ2 > 0
[removed] H0β2 ≥ 0; HAβ2 < 0

 

b-2. Find the p-value. (Round your answer to 4 decimal places.)

 

  p-value [removed]

 

b-3. At the 1% significance level, what is the conclusion to the test?
[removed] Do not reject H0Picture the unemployment rate and sales are not negatively related.
[removed] Do not reject H0Picture the unemployment rate and sales are negatively related.
[removed] Do not reject H0Picture the unemployment rate and sales are related.
[removed] Do not reject H0Picture the unemployment rate and sales are not related.

 

c-1. Choose the appropriate hypotheses to test whether advertising expenditures are positively related to sales.
[removed] H0β1 = 0; HAβ1 ≠ 0
[removed] H0β1 ≥ 0; HAβ1 < 0
[removed] H0β1 ≤ 0; HAβ1 > 0

 

c-2. Find the p-value. (Round your answer to 4 decimal places.)

 

  p-value [removed]

 

c-3. At the 1% significance level, what is the conclusion to the test?
[removed] Reject H0Picture advertising expenditures and sales are positively related.
[removed] Do not reject H0Picture advertising expenditures and sales are not positively related.
[removed] Do not reject H0Picture advertising expenditures and sales are positively related.
[removed] Reject H0Picture advertising expenditures and sales are not positively related.
3 –
For a sample of 20 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $1,000s). A portion of the regression results are as follows. Use Table 2 and Table 4.

 

  ANOVA df SS MS F Significance F
  Regression 2 2,576.7 1,288.4 ? 0.8163
  Residual 17 106,595.19 6,270.31
  Total 19 109,171.88

 

Coefficients Standard Error t Stat p-value Lower 95% Upper 95%
  Intercept 800.10 126.6195 6.3189 0.0000 532.95 1,067.24
  Poverty 0.5779 6.3784 0.0906 0.9289 −12.88 14.04
  Income −10.1429 16.1955 −0.6263 0.5395 −44.31 24.03

 

 a. Specify the sample regression equation. (Negative values should be indicated by a minus sign. Report your answers to 4 decimal places.)

 

  formula40.mml =[removed] + [removed] Poverty + [removed] Income

 

b-1. Choose the appropriate hypotheses to test whether the poverty rate and the crime rate are linearly related.
[removed] H0β1 ≥ 0; HAβ1 < 0
[removed] H0β1 ≤ 0; HAβ1 > 0
[removed] H0β1 = 0; HAβ1 ≠ 0

 

b-2. At the 5% significance level, what is the conclusion to the hypothesis test?
[removed] Do not reject H0Picture the poverty rate and the crime rate are not linearly related.
[removed] Reject H0Picture the poverty rate and the crime rate are linearly related.
[removed] Do not reject H0Picture the poverty rate and the crime rate are linearly related.
[removed] Reject H0Picture the poverty rate and the crime rate are not linearly related.

 

c-1. Construct a 95% confidence interval for the slope coefficient of income. (Negative values should be indicated by a minus sign. Round your intermediate calculations to 4 decimal places, “tα/2,df” value to 3 decimal places and final answers to 2 decimal places.)

 

  Confidence interval [removed] to [removed]

 

c-2. Using the confidence interval, determine whether income is significant in explaining the crime rate at the 5% significance level.
[removed] Income is significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero.
[removed] Income is not significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero.
[removed] Income is significant in explaining the crime rate, since its slope coefficient significantly differs from zero.
[removed] Income is not significant in explaining the crime rate, since its slope coefficient significantly differs from zero.

 

d-1. Choose the appropriate hypotheses to determine whether the poverty rate and income are jointly significant in explaining the crime rate.
[removed] H0β1 = β2 = 0; HA: At least one β j < 0
[removed] H0β1 = β2 = 0; HA: At least one β j ≠ 0
[removed] H0β1 = β2 = 0; HA: At least one β j > 0

 

d-2. At the 5% significance level, are the poverty rate and income jointly significant in explaining the crime rate?
[removed] No, since the null hypothesis is not rejected.
[removed] Yes, since the null hypothesis is rejected.
[removed] No, since the null hypothesis is rejected.
[removed] Yes, since the null hypothesis is not rejected.

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