study time in X hours
8
5
7
12
10
test grade Y in percent
68
50
62
92
80
a)model the date in a linear function
b)interpret the meaning of the slope
5 , 50
7 , 62
8 , 68
10 , 80
12 , 92
Different :
7 - 5 = 2
62 - 50 = 12
12 / 2 = 6
That mean :
y increases for 6 when x increases for 1
That is slope.
Slope intercept equation of straight line :
y = m ( x + x1 ) + y1
In this case :
m = ( y2 - y1 ) / ( x2 - x1 )
You can take any two points , example:
x1 = 8
y1 = 68
x2 = 10
y2 = 80
m = ( y2 - y1 ) / ( x2 - x1 )
m = ( 80 - 68 ) / ( 10 - 8 ) = 12 / 2 = 6
Slope = 6
y = m ( x - x1 ) + y1
y = 6 ( x - 8 ) + 68
y = 6 x - 48 + 68
y = 6 x + 20
Proof :
x = 8
y = 6 * 8 + 20 = 48 + 2 = 68
x = 5
y = 6 * 5 + 20 = 30 + 20 = 50
x = 7
y = 6 * 7 + 20 = 42 + 20 = 62
x = 12
y = 6 * 12 + 20 = 72 + 20 = 92
x = 10
y = 6 * 10 + 20 = 60 + 20 = 80
i need help on my math work
To model the data in a linear function, we can use the study time (X) as the input variable and the test grade (Y) as the output variable. This way, we can find a linear relationship between the two.
Let's list the values of X (study time in hours) and Y (test grade in percent):
X: 8, 5, 7, 12, 10
Y: 68, 50, 62, 92, 80
To create a linear function, we'll use the formula: Y = mx + b
where m is the slope of the line and b is the y-intercept.
Step 1: Find the slope (m):
To calculate the slope, we'll use the following formula:
m = (sum(X * Y) - (sum(X) * sum(Y)) / (n * sum(X^2) - (sum(X))^2)
where n is the number of data points.
First, calculate the necessary sums:
sum(X) = 8 + 5 + 7 + 12 + 10 = 42
sum(Y) = 68 + 50 + 62 + 92 + 80 = 352
sum(X * Y) = (8 * 68) + (5 * 50) + (7 * 62) + (12 * 92) + (10 * 80) = 1756
sum(X^2) = (8^2) + (5^2) + (7^2) + (12^2) + (10^2) = 378
Now, substitute these values into the slope formula:
m = (1756 - (42 * 352)) / (5 * 378 - 42^2)
m = (1756 - 14784) / (1890 - 1764)
m = -13028 / 126
m = -103.27 (rounded to two decimal places)
Step 2: Find the y-intercept (b):
To calculate the y-intercept, we'll use the formula:
b = (sum(Y) - m * sum(X)) / n
Substitute the known values:
b = (352 - (-103.27 * 42)) / 5
b = (352 + 4342.34) / 5
b = 4694.34 / 5
b = 938.87 (rounded to two decimal places)
Therefore, the linear function representing the relationship between the study time (X) and the test grade (Y) is:
Y = -103.27X + 938.87
Now, let's interpret the meaning of the slope (m):
The slope of the linear function is -103.27, which means that for every one-hour increase in study time, the test grade is expected to decrease by 103.27%. In other words, as the study time increases, the test grade tends to decrease at a rate of 103.27% per hour.