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Give step-by-step solution with explanation and final answer:Refer to Table 1. Table 1 contains the ACT scores and the GPA (grade point average) for eight college students. ACT tests were taken by students when they were in high school, to assess their readiness to attend college. GPA are students grades once in college: GPA i based on a four-point scale and has been rounded to one digt aftr the decimal. Consider the simple model: GPA = fo + BACT * +. (1). Using the data from Table 1, 3nd based on your OLS estimation of model (1) using Data Analysis tool in Excel, you can predict that an increase of an ACT score by 5, evaluated at the ACT = 20, will lead to an increase in college GPA by. 14GPAL. NOTE: In model (1), ACT s to the power %. Please write your answer in number format, with ONLY 1 decimal place of precision level: do not write your answer 3 a fraction. Add a leading minus sign symbol, a leading zero and trling zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. Table 1. Student Scores Student GPAACT 1 282 2 aa 3 302 4 as 5 362 6 3025 7 27s 8 aa

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Give step-by-step solution with explanation and final answer:Uploaded ImageRefer to Table 1. Table 1 contains the ACT scores and the GPA (grade point average) for eight college students. ACT tests were taken by students when they were in high school, to assess their readiness to attend college. GPA are students grades once in college: GPA i based on a four-point scale and has been rounded to one digt aftr the decimal. Consider the simple model: GPA = fo + BACT * +. (1). Using the data from Table 1, 3nd based on your OLS estimation of model (1) using Data Analysis tool in Excel, you can predict that an increase of an ACT score by 5, evaluated at the ACT = 20, will lead to an increase in college GPA by. 14GPAL. NOTE: In model (1), ACT s to the power %. Please write your answer in number format, with ONLY 1 decimal place of precision level: do not write your answer 3 a fraction. Add a leading minus sign symbol, a leading zero and trling zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. Table 1. Student Scores Student GPAACT 1 282 2 aa 3 302 4 as 5 362 6 3025 7 27s 8 aa

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Asked by: Nishant Saini
Created at: 2025-09-27 16:33:01

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Let's break down the problem step by step: ## **Step 1: Write the** The model: \[ \text{GPA = \beta + \_1 \ot \textACT}^{12} + u\] ## **Step : Prepare the Data** | Student GPA | ACT | ACT^{12} | ---------|----------|-----------| 1 2. | 21 | 4.5837 | | | .4 |24 |4.898 | | 3 | 3. | 26 | 5099 | | 4 | 3.5 | 27 | 5.1962 | | 5 | 3.6 | 29 | 5.3852 | | 6 | 3. | 25 | 5.000 | | 7 | 2.7 | 25 | 5.000 | | 8 | 3.7 | 30 | 5.4772 | (Square roots are rounded to 4 decimal places for calculation.) ## **Step 3: Estimate the Model (OLS Regression)** We need to run a simple linear regression of GPA on ACT^{1/2}. Formula for the slope (\(\beta_1\)): \[ \beta_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \] Where: - \(X_i = \text{ACT}^{1/2}\) - \(Y_i = \text{GPA}\) Let's compute the mean values: \[ \bar{X} = \frac{4.5837 + 4.89898 + 5.099 + 5.1962 + 5.3852 + 5.000 + 5.000 + 5.4772}{8} = 5.0808 \] \[ \bar{Y} = \frac{2.8 + 3.4 + 3. + 3.5 + 3.6 + 3. + 2.7 + 3.7}{8} = 3.2125 \] Now, let's calculate \(\sum (X_i - \bar{X})(Y_i - \bar{Y})\) and \(\sum (X_i - \bar{X})^2\): | Student | \(X_i\) | \(Y_i\) | \(X_i - \bar{X}\) | \(Y_i - \bar{Y}\) | Product | Square | |---------|--------|--------|------------------|------------------|---------|--------| | 1 | 4.5837 | 2.8 | -.4971 | -.4125 | .2051 | .2471 | | 2 | 4.899 | 3.4 | -.1818 | .1875 | -.0341 | .0331 | | 3 | 5.099 | 3. | .0182 | -.2125 | -.0039 | .0003 | | 4 | 5.1962 | 3.5 | .1154 | .2875 | .0332 | .0133 | | 5 | 5.3852 | 3.6 | .3044 | .3875 | .1179 | .0927 | | 6 | 5.000 | 3. | -.0808 | -.2125 | .0172 | .0065 | | 7 | 5.000 | 2.7 | -.0808 | -.5125 | .0414 | .0065 | | 8 | 5.4772 | 3.7 | .3964 | .4875 | .1932 | .1571 | Now sum the products and squares: \[ \sum \text{Product} = .2051 - .0341 - .0039 + .0332 + .1179 + .0172 + .0414 + .1932 = .5699 \] \[ \sum \text{Square} = .2471 + .0331 + .0003 + .0133 + .0927 + .0065 + .0065 + .1571 = .5566 \] So, \[ \beta_1 = \frac{.5699}{.5566} = 1.024 \] ## **Step 4: Calculate the Marginal Effect** The marginal effect at \(ACT = 20\) (when ACT increases by 5): The derivative: \[ \frac{d(\text{GPA})}{d(\text{ACT})} = \frac{1}{2} \beta_1 \cdot \text{ACT}^{-1/2} \] At \(ACT = 20\): \[ \frac{d(\text{GPA})}{d(\text{ACT})} = \frac{1}{2} \times 1.024 \times (20^{-1/2}) = .512 \times \frac{1}{\sqrt{20}} = .512 \times .2236 = .1144 \] For a 5-point increase: \[ \Delta \text{GPA} \approx .1144 \times 5 = .572 \] ## **Step 5: Final Answer** **An increase of 5 in ACT score, evaluated at ACT = 20, will lead to an increase in college GPA by:** \[ \boxed{.6} \] (Rounded to one decimal place, per instructions.)

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