Alex puts his spare change in a jar every night. If he has $11.09 at the end of January, $22.27 at the end of February, $44.35 in April, $75.82 in July, $89 in August, and $114.76 at the end of October, perform a linear regression on this data to complete the following items.
Note: I can't seem to perform a linear regression on my calculator correctly.
1. What does the value of the correlation coefficient tell you about the correlation of the data?
2. Write the equation of the best-fitting line. (Round to the nearest thousandths)
3. On average, how much money does Alex add to the jar each month?
4. Alex wants to buy a video game console at the end of December for $140. Will he have enough for this purchase? (Show work)
dunno about your calculator, but at
http://www.alcula.com/calculators/statistics/linear-regression/
we get
y = 11.33x - 0.90
what does the value of the correlation coefficient tell you about the correlation of the data ?
To perform a linear regression on the given data, we need to find the equation of the best-fitting line. Here are the steps to do so:
Step 1: Assign numerical values to the months. Let's use January as 1, February as 2, and so on.
Step 2: Create a table to represent the data as (x, y) pairs, where x represents the month and y represents the amount of money in the jar.
```
Month (x) Amount (y)
1 11.09
2 22.27
4 44.35
7 75.82
8 89
10 114.76
```
Step 3: Calculate the sum of x, y, x^2, and xy.
```
n = 6 (number of data points)
Sum of x = 1 + 2 + 4 + 7 + 8 + 10 = 32
Sum of y = 11.09 + 22.27 + 44.35 + 75.82 + 89 + 114.76 = 357.29
Sum of x^2 = 1^2 + 2^2 + 4^2 + 7^2 + 8^2 + 10^2 = 246
Sum of xy = (1 * 11.09) + (2 * 22.27) + (4 * 44.35) + (7 * 75.82) + (8 * 89) + (10 * 114.76) = 4714.27
```
Step 4: Calculate the slope (m) and the y-intercept (b) using the following formulas:
```
m = (n * Sum of xy - Sum of x * Sum of y) / (n * Sum of x^2 - (Sum of x)^2)
b = (Sum of y - m * Sum of x) / n
```
Substituting the values:
```
m = (6 * 4714.27 - 32 * 357.29) / (6 * 246 - 32^2) ≈ 16.221
b = (357.29 - 16.221 * 32) / 6 ≈ -1.310
```
The equation of the best-fitting line is y ≈ 16.221x - 1.31 (rounded to the nearest thousandths).
Now, let's address the questions:
1. The value of the correlation coefficient tells us about the correlation of the data. In linear regression, the correlation coefficient ranges from -1 to 1. If the correlation coefficient is close to -1, it indicates a strong negative correlation. If it's close to 1, it indicates a strong positive correlation. And if it's close to 0, it indicates no significant correlation or a weak correlation.
2. The equation of the best-fitting line is y ≈ 16.221x - 1.31.
3. To find out how much money Alex adds to the jar on average each month, we can use the slope of the line (m) obtained from the linear regression. In this case, the slope is approximately 16.221. This means Alex adds about $16.221 to the jar each month, on average.
4. To determine if Alex will have enough money to buy the video game console in December, let's calculate the jar's value at the end of December using the equation of the best-fitting line.
```
Month (x) Amount (y)
12 ?
```
Using the equation y ≈ 16.221x - 1.31, substitute x = 12:
y ≈ 16.221 * 12 - 1.31 ≈ 193.03
Since the jar will have approximately $193.03 at the end of December, Alex will have enough to buy the $140 video game console.