Math / Data & Statistics

QuestionListed below are the numbers of cricket chirps in 1 minute and the corresponding temperatures in ${ }^{\circ} \mathrm{F}$ . Find the regression equation, letting chirps in 1 minute be the independent ( $x$ ) variable. Find the best predicted temperature at a time when a cricket chirps 3000 times in 1 minute, using the regression equation. What is wrong with this predicted temperature? Use a significance level of 0.05 . \begin{tabular}{l|cccccccc} Chirps in 1 min & 973 & 752 & 1048 & 973 & 848 & 1071 & 846 & 1128 \\ \hline Temperature ( ${ }^{\circ} \mathrm{F}$ ) & 77.1 & 66 & 86.6 & 83.5 & 73.5 & 85.8 & 76.5 & 83.9 \end{tabular}
The regression equation is $\hat{y}=$ $\square$ $+$ $\square$ $x$ . (Round the $y$ -intercept to one decimal place as needed. Round the slope to four decimal places as needed.)

Studdy Solution

STEP 1

1. The relationship between the number of cricket chirps and temperature can be modeled using linear regression.
2. The independent variable $x$ is the number of chirps in 1 minute.
3. The dependent variable $y$ is the temperature in degrees Fahrenheit.
4. We are using a significance level of 0.05 to evaluate the regression model.

STEP 2

1. Calculate the regression equation.
2. Use the regression equation to predict the temperature for 3000 chirps.
3. Evaluate the prediction and discuss any potential issues.

STEP 3

Calculate the means of $x$ and $y$ .
$\bar{x} = \frac{973 + 752 + 1048 + 973 + 848 + 1071 + 846 + 1128}{8}$
$\bar{y} = \frac{77.1 + 66 + 86.6 + 83.5 + 73.5 + 85.8 + 76.5 + 83.9}{8}$

STEP 4

Calculate the slope $b_1$ using the formula:
$b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

STEP 5

Calculate the y-intercept $b_0$ using the formula:
$b_0 = \bar{y} - b_1 \bar{x}$

STEP 6

Substitute the values of $b_0$ and $b_1$ into the regression equation:
$\hat{y} = b_0 + b_1 x$

STEP 7

Predict the temperature when the number of chirps is 3000:
$\hat{y} = b_0 + b_1 \times 3000$

STEP 8

Discuss the prediction:
- Evaluate if the prediction is reasonable given the range of data. - Discuss any extrapolation concerns since 3000 chirps is outside the observed data range.
The regression equation is:
$\hat{y} = b_0 + b_1 x$
The predicted temperature for 3000 chirps is:
$\hat{y} = b_0 + b_1 \times 3000$
Note: The actual numerical values for $b_0$ and $b_1$ need to be calculated using the provided data.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the regression equation.
2. Use the regression equation to predict the temperature for 3000 chirps.
3. Evaluate the prediction and discuss any potential issues.

STEP 3

Calculate the means of $x$ and $y$ .
$\bar{x} = \frac{973 + 752 + 1048 + 973 + 848 + 1071 + 846 + 1128}{8}$
$\bar{y} = \frac{77.1 + 66 + 86.6 + 83.5 + 73.5 + 85.8 + 76.5 + 83.9}{8}$

STEP 4

Calculate the slope $b_1$ using the formula:
$b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

STEP 5

Calculate the y-intercept $b_0$ using the formula:
$b_0 = \bar{y} - b_1 \bar{x}$

STEP 6

Substitute the values of $b_0$ and $b_1$ into the regression equation:
$\hat{y} = b_0 + b_1 x$

STEP 7

Predict the temperature when the number of chirps is 3000:
$\hat{y} = b_0 + b_1 \times 3000$

STEP 8

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Studdy Solution

STEP 1

STEP 2

1. Calculate the regression equation.2. Use the regression equation to predict the temperature for 3000 chirps.3. Evaluate the prediction and discuss any potential issues.

STEP 3

STEP 4

Calculate the slope b1 b_1 b1​ using the formula:b1=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2 b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} b1​=∑(xi​−xˉ)2∑(xi​−xˉ)(yi​−yˉ​)​

STEP 5

Calculate the y-intercept b0 b_0 b0​ using the formula:b0=yˉ−b1xˉ b_0 = \bar{y} - b_1 \bar{x} b0​=yˉ​−b1​xˉ

STEP 6

Substitute the values of b0 b_0 b0​ and b1 b_1 b1​ into the regression equation:y^=b0+b1x \hat{y} = b_0 + b_1 x y^​=b0​+b1​x

STEP 7

Predict the temperature when the number of chirps is 3000:y^=b0+b1×3000 \hat{y} = b_0 + b_1 \times 3000 y^​=b0​+b1​×3000

STEP 8

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1. Calculate the regression equation.
2. Use the regression equation to predict the temperature for 3000 chirps.
3. Evaluate the prediction and discuss any potential issues.

Calculate the slope $b_1$ using the formula:
$b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

Calculate the y-intercept $b_0$ using the formula:
$b_0 = \bar{y} - b_1 \bar{x}$

Substitute the values of $b_0$ and $b_1$ into the regression equation:
$\hat{y} = b_0 + b_1 x$

Predict the temperature when the number of chirps is 3000:
$\hat{y} = b_0 + b_1 \times 3000$