A researcher has estimated a linear model to study the effect of weekly household income xi (in $100) on weekly household expenditure on food yi (in $). Using a sample of size N = 40, she found that
yb = 83.42 + 10.21x , R2 = 0.384 ii
(43.41)
and PNi=1 (yi − y ̄)2 = 500, 000 and the sample mean of xi is 19.605. Consider a
hypothesis testing against
H0 : the slope coefficient is 3.94. Therefore we, at 5% significance level,
(a) reject H0.
(b) do not reject H0.
(c) accept H1.
(d) cannot do anything unless the significance level is 10%.
(e) re-estimate the regression model using a different data.

To determine whether we reject or fail to reject $ H_0 $ (that the slope coefficient is $ 3.94 $) at the 5% significance level, we need to perform a hypothesis test for the slope coefficient $ \beta_1 $. The test statistic for this hypothesis test is given by:

\[
t = \frac{\hat{\beta}_1 - \beta_1}{SE(\hat{\beta}_1)}
\]

where:
- $ \hat{\beta}_1 = 10.21 $ (the estimated slope coefficient),
- $ \beta_1 = 3.94 $ (the hypothesized slope coefficient under $ H_0 $),
- $ SE(\hat{\beta}_1) = 43.41 $ (the standard error of the slope, given in parentheses).

### Step 1: Calculate the t-Statistic
Using the provided values:
\[
t = \frac{10.21 - 3.94}{43.41} = \frac{6.27}{43.41} \approx 0.1444
\]

### Step 2: Compare with the Critical Value
For a two-tailed test at the 5% significance level and $ N - 2 = 40 - 2 = 38 $ degrees of freedom, we can look up the critical $ t $-value for a 5% significance level (or use a table to approximate). For 38 degrees of freedom, the critical value for $ t $ is approximately $ \pm 2.024 $.

Since $ |t| = 0.1444 $ is far less than $ 2.024 $, we **do not reject $ H_0 $**.

### Conclusion
The correct answer is:
**(b) do not reject $ H_0 $**.

Would you like further details on hypothesis testing or interpreting regression outputs?

---

1. How would changing the significance level impact the decision?
2. How does the R-squared value affect the interpretation of this model?
3. Why is the standard error important in hypothesis testing?
4. What other factors could impact the reliability of this regression analysis?
5. How would a larger sample size affect the accuracy of our estimate?

**Tip:** In hypothesis testing, always compare the calculated t-statistic to the critical value based on your chosen significance level.

A researcher has estimated a linear model to study the effect of weekly household income xi (in $100) on weekly household expenditure on food yi (in $). Using a sample of size N = 40, she found that yb = 83.42 + 10.21x , R2 = 0.384 ii (43.41) and PNi=1 (yi − y ̄)2 = 500,000 and the sample mean of xi is 19.605. Consider a hypothesis testing against H0: the slope coefficient is 3.94. Therefore we, at 5% significance level, (a) reject H0. (b) do not reject H0. (c) accept H1. (d) cannot do anything unless the significance level is 10%. (e) re-estimate the regression model using a different data.

Explore a hypothesis test for the slope coefficient in a linear regression model estimating the effect of household income on food expenditure. This example uses a 5% significance level with sample data (N = 40) to test if the slope coefficient equals 3.94. Key statistical concepts include hypothesis testing, t-statistics, and regression interpretation.

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