John sells frozen fruit bars at a stand in a park during the summer months. He records the average weekly temperature and number of frozen fruit bars sold for 6 weeks.
| Temperature (°F) | Fruit Bars Sold |
|---|---|
| 67 | 50 |
| 71 | 54 |
| 76 | 63 |
| 76 | 65 |
| 82 | 65 |
| 87 | 100 |
What type of correlation exists between the temperature and the number of fruit bars sold?
What is the real-world meaning of the slope of the line of best fit for the given scenario?
There are approximately ___ more fruit bars sold for every ___ degree(s) the temperature rises.
Answer (2)
There is a positive correlation between temperature and the number of fruit bars sold.
Calculate the line of best fit for the data: y = m x + b .
The slope of the line of best fit is approximately m ≈ 2.19 .
For every 1 degree Fahrenheit increase in temperature, approximately 2 more fruit bars are sold. 2
Explanation
Analyzing the Data We are given a set of data points representing the average weekly temperature and the number of frozen fruit bars sold. Our goal is to determine the type of correlation between these two variables and to understand the real-world meaning of the slope of the line of best fit. The data points are (67, 50), (71, 54), (76, 63), (76, 65), (82, 65), and (87, 100).
Determining the Correlation Type First, let's determine the type of correlation. As the temperature increases, the number of fruit bars sold generally increases as well. This indicates a positive correlation between temperature and the number of fruit bars sold.
Finding the Line of Best Fit Next, we need to find the line of best fit for the data. The equation of the line of best fit is in the form y = m x + b , where y is the number of fruit bars sold, x is the temperature, m is the slope, and b is the y-intercept. Using the given data, we can calculate the slope and y-intercept. The result of this calculation gives us m ≈ 2.19 and b ≈ − 101.02 . Therefore, the line of best fit is approximately y = 2.19 x − 101.02 .
Interpreting the Slope Now, let's interpret the slope in the context of the problem. The slope, m ≈ 2.19 , represents the change in the number of fruit bars sold for every one-degree increase in temperature. In simpler terms, for every 1 degree Fahrenheit increase in temperature, approximately 2.19 more fruit bars are sold. We can round this to approximately 2 fruit bars.
Final Answer Therefore, there is a positive correlation between the temperature and the number of fruit bars sold. For every 1 degree Fahrenheit increase in temperature, approximately 2 more fruit bars are sold.
Examples
Understanding the correlation between temperature and sales can help John predict how many fruit bars he might sell on a given day based on the weather forecast. This allows him to stock his stand appropriately, minimizing waste and maximizing profit. For example, if the forecast predicts a hot day, John knows he needs to bring extra fruit bars to meet the expected demand. This type of analysis is also useful for planning staffing levels and ordering supplies.
There is a positive correlation between temperature and the number of fruit bars sold, with approximately 2 more fruit bars sold for every 1 degree Fahrenheit increase in temperature. This correlation helps John predict sales based on weather conditions. Understanding the relationship aids in effectively managing inventory and maximizing profits.
;
