Instructions: Use technology to create a scatter plot for the data below. Then find the answers to the following questions. (Note: You will not turn in the scatter plot for this question.)
| Coffees Sold | 30 | 29 | 26 | 24 | 19 | 20 | 15 | 13 | 12 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Daily Temperature (F) | 20 | 27 | 33 | 40 | 49 | 50 | 61 | 64 | 67 | 78 | 80 |
Answer (2)
Create a scatter plot to visualize the relationship between 'Daily Temperature (F)' and 'Coffees Sold'.
Calculate the correlation coefficient: r = − 0.9948 , indicating a strong negative correlation.
Determine the regression line equation: y = − 0.3869 x + 38.6499 .
Predict coffee sales for a temperature of 55°F: y = − 0.3869 ( 55 ) + 38.6499 = 17.3704 , so the predicted number of coffees sold is approximately 17.37 .
Explanation
Understanding the Problem We are given data for 'Coffees Sold' and 'Daily Temperature (F)' and asked to analyze the relationship between these two variables using a scatter plot. We need to determine the strength and direction of the relationship, and if a linear model is appropriate, find the equation of the regression line and use it for prediction.
Creating the Scatter Plot First, we create a scatter plot with 'Daily Temperature (F)' on the x-axis and 'Coffees Sold' on the y-axis. The problem states that we use technology to do this, but we don't need to submit the plot.
Analyzing the Scatter Plot Next, we analyze the scatter plot to determine the type of relationship between the two variables. Based on the plot (which we are not submitting), we can observe a strong negative linear relationship, meaning as the temperature increases, the number of coffees sold decreases.
Calculating the Correlation Coefficient Since a linear relationship is observed, we can calculate the correlation coefficient (r) to quantify the strength and direction of the linear relationship. The result of this calculation is r = − 0.9948 . This indicates a very strong negative correlation.
Finding the Regression Line Equation Now, we determine the equation of the least-squares regression line (y = a + bx), where y is 'Coffees Sold' and x is 'Daily Temperature (F)'. From the calculations, we find the slope (b) to be approximately -0.3869 and the intercept (a) to be approximately 38.6499. Therefore, the equation of the regression line is: y = − 0.3869 x + 38.6499
Predicting Coffee Sales We can use this equation to predict 'Coffees Sold' for a given 'Daily Temperature (F)'. For example, if the daily temperature is 55°F, the predicted number of coffees sold is: y = − 0.3869 ( 55 ) + 38.6499 = 17.3704
Assessing Goodness of Fit To assess the goodness of fit of the regression line, we calculate the coefficient of determination ( r 2 ). The result of the calculation is r 2 = 0.9897 . This means that approximately 98.97% of the variation in 'Coffees Sold' can be explained by the variation in 'Daily Temperature (F)'.
Final Answer Therefore, based on the calculations, the predicted number of coffees sold when the temperature is 55°F is approximately 17.37.
Examples
Understanding the relationship between temperature and coffee sales can help a coffee shop owner optimize their inventory and staffing. For instance, on hotter days, they might reduce their coffee stock and increase their cold beverage options. Conversely, on colder days, they can prepare for higher coffee demand. This data-driven approach ensures efficient resource allocation and maximizes profit.
The relationship between 'Coffees Sold' and 'Daily Temperature (F)' can be analyzed through a scatter plot, which typically shows a negative correlation, indicating that as temperatures rise, coffee sales decrease. By calculating the correlation coefficient and regression equation, we can predict coffee sales for specific temperatures. This data helps optimize business operations in a coffee shop.
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