Question

the average driving distance (yards) and driving accuracy (percent of drives that land in the fairway) for 8 golfers are recorded in the table to the right. complete parts a through e below.

player	distance (yards)	accuracy (%)
2	303.7	55.7
3	309.2	49.8
4	313.4	47.2
5	294.2	61.3
6	291.1	65.8
7	286.5	59.0
8	301.7	56.1

a. write the equation of a straight - line model relating driving accuracy (y) to driving distance (x). choose the correct answer below.

a. $y = \beta_0+\beta_1x + \epsilon$

b. $y = \beta_1x$

c. $y = \beta_1x^2+\beta_0$

d. $y = \beta_1x+\epsilon$

b. fit the model, part a, to the data using simple linear regression. give the least squares prediction equation.

$\hat{y}=\square + \square x$

Explanation:

Part a

A straight - line (linear) model relating a response variable \(y\) (driving accuracy) to a predictor variable \(x\) (driving distance) in the context of regression analysis has the form \(y=\beta_0+\beta_1x + \epsilon\), where \(\beta_0\) is the y - intercept, \(\beta_1\) is the slope, and \(\epsilon\) is the error term. Option B is a simple proportionality without an intercept and error term, option C is a quadratic model (not linear), and option D is missing the intercept term. So the correct model is the one with the intercept, slope, and error term.

Step 1: Calculate necessary sums

We calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^{2}\) from the given data points.

Step 2: Calculate the slope \(\hat{\beta}_1\)

Use the formula \(\hat{\beta}_1=\frac{n\sum xy - \sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\) with \(n = 8\).

Step 3: Calculate the intercept \(\hat{\beta}_0\)

Use the formula \(\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}\) and \(\bar{y}=\frac{\sum y}{n}\).

Answer:

A. \(y = \beta_0+\beta_1x+\epsilon\)

Part b

To find the least - squares prediction equation \(\hat{y}=\hat{\beta}_0+\hat{\beta}_1x\), we first need to calculate the necessary sums. Let \(x\) be the driving distance (yards) and \(y\) be the driving accuracy (%).

The data points are:
\((x_1,y_1)=(316.5,45.7)\), \((x_2,y_2)=(303.7,55.7)\), \((x_3,y_3)=(309.2,49.8)\), \((x_4,y_4)=(313.4,47.2)\), \((x_5,y_5)=(294.2,61.3)\), \((x_6,y_6)=(291.1,65.8)\), \((x_7,y_7)=(296.5,59.0)\), \((x_8,y_8)=(301.7,56.1)\)

First, calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^{2}\):

1. \(\sum x=316.5 + 303.7+309.2+313.4+294.2+291.1+296.5+301.7\)

$$
LATEXBLOCK0
$$

1. \(\sum y=45.7 + 55.7+49.8+47.2+61.3+65.8+59.0+56.1\)

$$
LATEXBLOCK1
$$

1. \(\sum xy=(316.5\times45.7)+(303.7\times55.7)+(309.2\times49.8)+(313.4\times47.2)+(294.2\times61.3)+(291.1\times65.8)+(296.5\times59.0)+(301.7\times56.1)\)

$$
LATEXBLOCK2
$$

$$
LATEXBLOCK3
$$

1. \(\sum x^{2}=316.5^{2}+303.7^{2}+309.2^{2}+313.4^{2}+294.2^{2}+291.1^{2}+296.5^{2}+301.7^{2}\)

$$
LATEXBLOCK4
$$

$$
LATEXBLOCK5
$$

The formula for the slope \(\hat{\beta}_1\) is:

$$ \hat{\beta}_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} $$

where \(n = 8\)

$$
LATEXBLOCK6
$$

$$
LATEXBLOCK7
$$

The formula for the intercept \(\hat{\beta}_0\) is:

$$ \hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x} $$

where \(\bar{x}=\frac{\sum x}{n}=\frac{2426.3}{8}=303.2875\) and \(\bar{y}=\frac{\sum y}{n}=\frac{440.6}{8}=55.075\)

$$
LATEXBLOCK8
$$

So the least - squares prediction equation is \(\hat{y}=524.05-1.543x\) (rounded to two decimal places for the coefficients).