Math / Data & Statistics

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The table below includes data from taxi rides. The distances are in miles, the times are in minutes, the fares are in dollars, and the tips are in dollars. Is there sufficient evidence to conclude that there is a linear correlation between the time of the ride and the tip amount? Construct a scatterplot, find the value of the linear correlation coefficient $r$ , and find the $P$ -value of $r$ . Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Use a significance level of $\alpha=0.01$ . Does it appear that riders base their tips on the time of the ride?
Click here for information on the taxi rides.
Construct a scatterplot. Choose the correct graph A.
Determine the linear correlation coefficient. The linear correlation coefficient is $\mathrm{r}=$ $\square$ (Round to three decimal places as needed.)
Taxi data \begin{tabular}{|lllllllllll|} \hline Distance & 0.67 & 2.51 & 8.48 & 12.73 & 1.65 & 1.02 & 1.32 & 0.49 & - \\ \hline Time & 6.00 & 18.00 & 31.00 & 27.00 & 11.00 & 8.00 & 8.00 & 2.00 & \\ Fare & 6.30 & 14.30 & 31.78 & 36.80 & 9.80 & 7.80 & 7.80 & 4.80 \\ Tip & 1.89 & 4.29 & 2.98 & 0.00 & 1.96 & 2.34 & 0.00 & 0.00 \\ \hline \end{tabular} Print Done

Studdy Solution

STEP 1

1. We have data for taxi rides including time and tip amounts.
2. We need to determine if there is a linear correlation between the time of the ride and the tip amount.
3. The significance level for testing the correlation is $\alpha = 0.01$ .

STEP 2

1. Construct a scatterplot of the data.
2. Calculate the linear correlation coefficient $r$ .
3. Determine the $P$ -value for the correlation coefficient.
4. Evaluate the evidence against the significance level to determine if there is a significant linear correlation.

STEP 3

Plot the data points on a scatterplot with 'Time' on the x-axis and 'Tip' on the y-axis. Each data point represents a pair $(\text{Time}, \text{Tip})$ .
Data points: - (6.00, 1.89) - (18.00, 4.29) - (31.00, 2.98) - (27.00, 0.00) - (11.00, 1.96) - (8.00, 2.34) - (8.00, 0.00) - (2.00, 0.00)

STEP 4

Calculate the linear correlation coefficient $r$ using the formula:
$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$
Where $x$ represents 'Time' and $y$ represents 'Tip'. Compute the necessary sums and substitute them into the formula.

STEP 5

Compute the following: - $\sum x = 6 + 18 + 31 + 27 + 11 + 8 + 8 + 2 = 111$ - $\sum y = 1.89 + 4.29 + 2.98 + 0.00 + 1.96 + 2.34 + 0.00 + 0.00 = 13.46$ - $\sum xy = 6 \times 1.89 + 18 \times 4.29 + 31 \times 2.98 + 27 \times 0.00 + 11 \times 1.96 + 8 \times 2.34 + 8 \times 0.00 + 2 \times 0.00 = 157.54$ - $\sum x^2 = 6^2 + 18^2 + 31^2 + 27^2 + 11^2 + 8^2 + 8^2 + 2^2 = 2267$ - $\sum y^2 = 1.89^2 + 4.29^2 + 2.98^2 + 0.00^2 + 1.96^2 + 2.34^2 + 0.00^2 + 0.00^2 = 35.57$
Substitute these into the formula for $r$ .

STEP 6

Substitute the computed values into the formula for $r$ :
$r = \frac{8(157.54) - (111)(13.46)}{\sqrt{[8 \times 2267 - 111^2][8 \times 35.57 - 13.46^2]}}$
Calculate $r$ .

STEP 7

Determine the $P$ -value for the calculated $r$ using statistical software or a correlation table for $n = 8$ .

STEP 8

Compare the $P$ -value with the significance level $\alpha = 0.01$ . If $P$ -value < $\alpha$ , there is sufficient evidence to conclude a linear correlation.
The linear correlation coefficient $r$ is approximately $\square$ .
Based on the $P$ -value, determine if there is a significant linear correlation between the time of the ride and the tip amount.

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Studdy Solution

STEP 1

1. We have data for taxi rides including time and tip amounts.
2. We need to determine if there is a linear correlation between the time of the ride and the tip amount.
3. The significance level for testing the correlation is $\alpha = 0.01$ .

STEP 2

1. Construct a scatterplot of the data.
2. Calculate the linear correlation coefficient $r$ .
3. Determine the $P$ -value for the correlation coefficient.
4. Evaluate the evidence against the significance level to determine if there is a significant linear correlation.

STEP 3

Plot the data points on a scatterplot with 'Time' on the x-axis and 'Tip' on the y-axis. Each data point represents a pair $(\text{Time}, \text{Tip})$ .
Data points: - (6.00, 1.89) - (18.00, 4.29) - (31.00, 2.98) - (27.00, 0.00) - (11.00, 1.96) - (8.00, 2.34) - (8.00, 0.00) - (2.00, 0.00)

STEP 4

Calculate the linear correlation coefficient $r$ using the formula:
$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$
Where $x$ represents 'Time' and $y$ represents 'Tip'. Compute the necessary sums and substitute them into the formula.

STEP 5

STEP 6

Substitute the computed values into the formula for $r$ :
$r = \frac{8(157.54) - (111)(13.46)}{\sqrt{[8 \times 2267 - 111^2][8 \times 35.57 - 13.46^2]}}$
Calculate $r$ .

STEP 7

Determine the $P$ -value for the calculated $r$ using statistical software or a correlation table for $n = 8$ .

STEP 8

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Studdy Solution

STEP 1

1. We have data for taxi rides including time and tip amounts.2. We need to determine if there is a linear correlation between the time of the ride and the tip amount.3. The significance level for testing the correlation is α=0.01\alpha = 0.01α=0.01.

STEP 2

1. Construct a scatterplot of the data.2. Calculate the linear correlation coefficient rrr.3. Determine the PPP-value for the correlation coefficient.4. Evaluate the evidence against the significance level to determine if there is a significant linear correlation.

STEP 3

STEP 4

STEP 5

STEP 6

Substitute the computed values into the formula for rrr:r=8(157.54)−(111)(13.46)[8×2267−1112][8×35.57−13.462]r = \frac{8(157.54) - (111)(13.46)}{\sqrt{[8 \times 2267 - 111^2][8 \times 35.57 - 13.46^2]}}r=[8×2267−1112][8×35.57−13.462]​8(157.54)−(111)(13.46)​Calculate rrr.

STEP 7

Determine the PPP-value for the calculated rrr using statistical software or a correlation table for n=8n = 8n=8.

STEP 8

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

1. We have data for taxi rides including time and tip amounts.
2. We need to determine if there is a linear correlation between the time of the ride and the tip amount.
3. The significance level for testing the correlation is $\alpha = 0.01$ .

1. Construct a scatterplot of the data.
2. Calculate the linear correlation coefficient $r$ .
3. Determine the $P$ -value for the correlation coefficient.
4. Evaluate the evidence against the significance level to determine if there is a significant linear correlation.

Substitute the computed values into the formula for $r$ :
$r = \frac{8(157.54) - (111)(13.46)}{\sqrt{[8 \times 2267 - 111^2][8 \times 35.57 - 13.46^2]}}$
Calculate $r$ .

Determine the $P$ -value for the calculated $r$ using statistical software or a correlation table for $n = 8$ .