Questions: Question 15 (1 point) A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let x represent the third homework score and y the midterm exam score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed. HW3 Midterm 13.3 59.811 21.9 87.539 9.7 53.728 5.4 39.174 20.9 89.729 18.5 78.985 15.4 73.274 25 93.25 9.7 52.257 6.4 43.984 20.2 79.762 21.8 84.258 23.1 92.911 23 87.82 11.4 45.034 14.9 71.869 18.4 76.704 15.1 60.431 15 65.15 16.8 77.208 Find the predicted midterm score when the homework 3 score is 19.5. Do not round until the end, then round answer to 2 decimal places.

Question 15 (1 point) A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let x represent the third homework score and y the midterm exam score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed. HW3 Midterm 13.3 59.811 21.9 87.539 9.7 53.728 5.4 39.174 20.9 89.729 18.5 78.985 15.4 73.274 25 93.25 9.7 52.257 6.4 43.984 20.2 79.762 21.8 84.258 23.1 92.911 23 87.82 11.4 45.034 14.9 71.869 18.4 76.704 15.1 60.431 15 65.15 16.8 77.208 Find the predicted midterm score when the homework 3 score is 19.5. Do not round until the end, then round answer to 2 decimal places.
Transcript text: Question 15 (1 point) A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let $x$ represent the third homework score and $y$ the midterm exam score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed. HW3 Midterm $13.3 \quad 59.811$ $21.9 \quad 87.539$ $9.7 \quad 53.728$ $5.4 \quad 39.174$ $20.9 \quad 89.729$ $18.5 \quad 78.985$ $15.4 \quad 73.274$ $25 \quad 93.25$ $9.7 \quad 52.257$ $6.4 \quad 43.984$ $20.2 \quad 79.762$ $21.8 \quad 84.258$ $23.1 \quad 92.911$ $23 \quad 87.82$ $11.4 \quad 45.034$ $14.9 \quad 71.869$ $18.4 \quad 76.704$ $15.1 \quad 60.431$ $15 \quad 65.15$ $16.8 \quad 77.208$ Find the predicted midterm score when the homework 3 score is 19.5 . Do not round until the end, then round answer to 2 decimal places.
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Solution

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Solution Steps

Step 1: Calculate the Means

The means of the homework scores \( \bar{x} \) and midterm scores \( \bar{y} \) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 16.295 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 70.6439 \]

Step 2: Calculate the Correlation Coefficient

The correlation coefficient \( r \) is computed to assess the strength of the linear relationship between the two variables:

\[ r = 0.9692 \]

Step 3: Calculate the Slope (β)

The slope \( \beta \) is determined using the following formulas:

Numerator for \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 24826.473 - 20 \cdot 16.295 \cdot 70.6439 = 1803.626 \]

Denominator for \( \beta \):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 5927.69 - 20 \cdot 16.295^2 = 617.1495 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{1803.626}{617.1495} = 2.9225 \]

Step 4: Calculate the Intercept (α)

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 70.6439 - 2.9225 \cdot 16.295 = 23.0216 \]

Step 5: Formulate the Equation of the Line of Best Fit

The equation of the line of best fit is given by:

\[ y = 23.0216 + 2.9225x \]

Step 6: Predict the Midterm Score for HW3 Score of 19.5

To find the predicted midterm score when the homework 3 score \( x = 19.5 \):

\[ \text{Predicted } y = \alpha + \beta \cdot 19.5 = 23.0216 + 2.9225 \cdot 19.5 = 80.01 \]

Final Answer

The predicted midterm score for a homework 3 score of 19.5 is:

\[ \boxed{80.01} \]

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