Math / Data & Statistics

QuestionUse the data in the table below to complete parts (a) through (d). \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline $\mathbf{x}$ & 39 & 33 & 41 & 47 & 42 & 50 & 59 & 54 & 53 \\ \hline $\mathbf{y}$ & 22 & 20 & 25 & 31 & 28 & 29 & 27 & 23 & 26 \\ \hline \end{tabular}
Click the icon to view details on how to construct and interpret residual plots. (a) Find the equation of the regression line. $\hat{y}=$ $\square$ $x+$ $\square$ (Round to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We've got some data, and we need to find the line of best fit that describes how $y$ changes with $x$ , and write out its equation. Watch out! Don't mix up $x$ and $y$ values!
Also, keep track of those decimals – accuracy is key!

STEP 2

1. Calculate the means of $x$ and $y$ .
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write out the equation.

STEP 3

Alright, let's add up all the $x$ values: $39 + 33 + 41 + 47 + 42 + 50 + 59 + 54 + 53 = 418$ .
Now, divide by the number of values, which is 9, to get the mean of $x$ : $\bar{x} = \frac{418}{9} \approx 46.444$ .

STEP 4

Let's do the same for $y$ ! Sum all the $y$ values: $22 + 20 + 25 + 31 + 28 + 29 + 27 + 23 + 26 = 221$ . Divide by 9 to get the mean of $y$ : $\bar{y} = \frac{221}{9} \approx 25.667$ .

STEP 5

The slope ( $b$ ) is calculated as: $b = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}.$ This might look scary, but it's just a fancy way of saying how much $y$ changes on average for every change in $x$ .

STEP 6

We need to calculate $(x_i - \bar{x})(y_i - \bar{y})$ for each pair of $x$ and $y$ and then sum them up.
This gives us approximately $258.222$ .

STEP 7

Now, we calculate $(x_i - \bar{x})^2$ for each $x$ and sum them up.
This gives us approximately $568.889$ .

STEP 8

Now, divide the numerator by the denominator: $b = \frac{258.222}{568.889} \approx 0.454$ .
So, for every increase of 1 in $x$ , $y$ increases by roughly 0.454!

STEP 9

The y-intercept ( $a$ ) is where the line crosses the y-axis when $x = 0$ .
It's calculated as: $a = \bar{y} - b\bar{x}.$

STEP 10

Plug in the values we found: $a = 25.667 - (0.454)(46.444) \approx 4.628$ .

STEP 11

The equation of the regression line is $\hat{y} = a + bx$ .
Plugging in our slope and y-intercept, we get $\hat{y} = 4.628 + 0.454x$ .

STEP 12

The equation of the regression line is $\hat{y} = 0.454x + 4.628$ .

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We've got some data, and we need to find the line of best fit that describes how $y$ changes with $x$ , and write out its equation. Watch out! Don't mix up $x$ and $y$ values!
Also, keep track of those decimals – accuracy is key!

STEP 2

1. Calculate the means of $x$ and $y$ .
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write out the equation.

STEP 3

Alright, let's add up all the $x$ values: $39 + 33 + 41 + 47 + 42 + 50 + 59 + 54 + 53 = 418$ .
Now, divide by the number of values, which is 9, to get the mean of $x$ : $\bar{x} = \frac{418}{9} \approx 46.444$ .

STEP 4

Let's do the same for $y$ ! Sum all the $y$ values: $22 + 20 + 25 + 31 + 28 + 29 + 27 + 23 + 26 = 221$ . Divide by 9 to get the mean of $y$ : $\bar{y} = \frac{221}{9} \approx 25.667$ .

STEP 5

The slope ( $b$ ) is calculated as: $b = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}.$ This might look scary, but it's just a fancy way of saying how much $y$ changes on average for every change in $x$ .

STEP 6

We need to calculate $(x_i - \bar{x})(y_i - \bar{y})$ for each pair of $x$ and $y$ and then sum them up.
This gives us approximately $258.222$ .

STEP 7

Now, we calculate $(x_i - \bar{x})^2$ for each $x$ and sum them up.
This gives us approximately $568.889$ .

STEP 8

Now, divide the numerator by the denominator: $b = \frac{258.222}{568.889} \approx 0.454$ .
So, for every increase of 1 in $x$ , $y$ increases by roughly 0.454!

STEP 9

The y-intercept ( $a$ ) is where the line crosses the y-axis when $x = 0$ .
It's calculated as: $a = \bar{y} - b\bar{x}.$

STEP 10

Plug in the values we found: $a = 25.667 - (0.454)(46.444) \approx 4.628$ .

STEP 11

The equation of the regression line is $\hat{y} = a + bx$ .
Plugging in our slope and y-intercept, we get $\hat{y} = 4.628 + 0.454x$ .

STEP 12

The equation of the regression line is $\hat{y} = 0.454x + 4.628$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? We've got some data, and we need to find the *line of best fit* that describes how yyy changes with xxx, and write out its equation. Watch out! Don't mix up xxx and yyy values!Also, keep track of those decimals – accuracy is key!

STEP 2

1. Calculate the means of xxx and yyy.2. Calculate the slope of the regression line.3. Calculate the y-intercept of the regression line.4. Write out the equation.

STEP 3

STEP 4

Let's do the same for yyy! **Sum** all the yyy values: 22+20+25+31+28+29+27+23+26=22122 + 20 + 25 + 31 + 28 + 29 + 27 + 23 + 26 = 22122+20+25+31+28+29+27+23+26=221. **Divide** by **9** to get the **mean** of yyy: yˉ=2219≈25.667\bar{y} = \frac{221}{9} \approx 25.667yˉ​=9221​≈25.667.

STEP 5

STEP 6

We need to calculate (xi−xˉ)(yi−yˉ)(x_i - \bar{x})(y_i - \bar{y})(xi​−xˉ)(yi​−yˉ​) for each pair of xxx and yyy and then **sum** them up.This gives us approximately 258.222258.222258.222.

STEP 7

Now, we calculate (xi−xˉ)2(x_i - \bar{x})^2(xi​−xˉ)2 for each xxx and **sum** them up.This gives us approximately 568.889568.889568.889.

STEP 8

Now, **divide** the numerator by the denominator: b=258.222568.889≈0.454b = \frac{258.222}{568.889} \approx 0.454b=568.889258.222​≈0.454.So, for every increase of 1 in xxx, yyy increases by roughly **0.454**!

STEP 9

The **y-intercept** (aaa) is where the line crosses the y-axis when x=0x = 0x=0.It's calculated as: a=yˉ−bxˉ.a = \bar{y} - b\bar{x}.a=yˉ​−bxˉ.

STEP 10

Plug in the values we found: a=25.667−(0.454)(46.444)≈4.628a = 25.667 - (0.454)(46.444) \approx 4.628a=25.667−(0.454)(46.444)≈4.628.

STEP 11

The equation of the regression line is y^=a+bx\hat{y} = a + bxy^​=a+bx.Plugging in our **slope** and **y-intercept**, we get y^=4.628+0.454x\hat{y} = 4.628 + 0.454xy^​=4.628+0.454x.

STEP 12

The equation of the regression line is y^=0.454x+4.628\hat{y} = 0.454x + 4.628y^​=0.454x+4.628.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We've got some data, and we need to find the line of best fit that describes how $y$ changes with $x$ , and write out its equation. Watch out! Don't mix up $x$ and $y$ values!
Also, keep track of those decimals – accuracy is key!

1. Calculate the means of $x$ and $y$ .
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write out the equation.

Let's do the same for $y$ ! Sum all the $y$ values: $22 + 20 + 25 + 31 + 28 + 29 + 27 + 23 + 26 = 221$ . Divide by 9 to get the mean of $y$ : $\bar{y} = \frac{221}{9} \approx 25.667$ .

We need to calculate $(x_i - \bar{x})(y_i - \bar{y})$ for each pair of $x$ and $y$ and then sum them up.
This gives us approximately $258.222$ .

Now, we calculate $(x_i - \bar{x})^2$ for each $x$ and sum them up.
This gives us approximately $568.889$ .

Now, divide the numerator by the denominator: $b = \frac{258.222}{568.889} \approx 0.454$ .
So, for every increase of 1 in $x$ , $y$ increases by roughly 0.454!

The y-intercept ( $a$ ) is where the line crosses the y-axis when $x = 0$ .
It's calculated as: $a = \bar{y} - b\bar{x}.$

Plug in the values we found: $a = 25.667 - (0.454)(46.444) \approx 4.628$ .

The equation of the regression line is $\hat{y} = a + bx$ .
Plugging in our slope and y-intercept, we get $\hat{y} = 4.628 + 0.454x$ .

The equation of the regression line is $\hat{y} = 0.454x + 4.628$ .