1: Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
Scenario 1:
Two friends are selling lemonade on a hot summer day. The price they charge varies depending on the number of cups they sell.
Data pairs:
- If they sell 2 cups, they charge $1.50.
- If they sell 4 cups, they charge $3.00.
- If they sell 6 cups, they charge $4.50.
- If they sell 8 cups, they charge $6.00.
Scenario 2:
A car is driving down the highway and the rate at which it consumes fuel depends on its speed.
Data pairs:
- If the car is going 30 mph, it uses 2 gallons of fuel per hour.
- If the car is going 40 mph, it uses 2.5 gallons of fuel per hour.
- If the car is going 50 mph, it uses 3 gallons of fuel per hour.
- If the car is going 60 mph, it uses 3.5 gallons of fuel per hour.
Of the mean, median, and mode, which measure of center is most affected by outliers? Explain your reasoning including an example with at least 10 data points.
The measure of center that is most affected by outliers is the mean.
The mean is the sum of all data points divided by the number of data points, which makes it sensitive to the value of each individual data point. This means that if there are outliers, which are data points that are much larger or smaller than the other data points, they can have a significant impact on the mean.
For example, consider the following set of 10 data points:
10, 15, 20, 25, 30, 35, 40, 45, 50, 100
The mean of this set is 33.
However, if we add an outlier to this set, it can dramatically change the mean. Let's say we add a value of 500 to the set:
10, 15, 20, 25, 30, 35, 40, 45, 50, 500
Now the mean of this set is 98.5, which is much higher than the original mean. This is because the outlier value of 500 has a much larger impact on the sum of all the data values, pulling the mean towards it.
In contrast, the median and mode are less affected by outliers. The median is the middle value when data is arranged in order, so outliers have less impact because they don't significantly affect the position of the middle value. The mode is the most frequent value in the data set, so it is not affected by outliers that occur infrequently.
Scenario 1: Direct Variation
In this scenario, let's consider the relationship between the number of hours (x) a car is driven and the distance (y) traveled by the car in miles.
Data pairs:
1. (2, 40)
2. (3, 60)
3. (4, 80)
4. (5, 100)
Scenario 2: Inverse Variation
In this scenario, let's consider the relationship between the amount of water (x) poured into a container and the time (y) it takes for the container to drain completely in seconds.
Data pairs:
1. (10, 4)
2. (20, 2)
3. (30, 1.33)
4. (40, 1)
Please note that Scenario 1 represents direct variation, while Scenario 2 represents inverse variation. It is up to your classmates to determine the value of k in each situation.