Download Ch. 4 SOL Review: Variation/Statistics AII.10 Variation I Direct

Ch. 4 SOL Review: Variation/Statistics
AII.10 Variation
I Direct Variation
 Equation: y = kx (k is the constant of variation);
 graph is a line thru the origin
 Solve base equation for k and substitute it and remaining numbers into eqn. again)
Example 1: If y varies directly as x and y is 6 when x is 18, find y when x is 24.
2) Then plug k into the formula and
y  kx
find the missing variable:
1) Set up to solve for k: 6  18k
y  kx
1
k
1
y  (24)
3
3
y 8
II Inverse Variation
 Example – the speed of a car and the time it takes to reach the destination
k
 Equation: y  ( k is the constant variation)
x
 graph is a hyperbola in opposite quadrants (Quad I & III or Quad II and IV)
 To solve find k and substitute it and remaining numbers into eqn. again.
Example 2:
If y varies inversely as x and y = 10 when x = 2, find y when x = 6.
k
y
k
x
y
k
x
If 10 
Find y now
20 10
2
y

20  k
6
3
Example 3:
The length of a violin string varies inversely as the frequency of its vibrations. A violin
string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of
an 8-inch string. (Use l for length and f for frequency.)
III Joint
 A combination of direct and inverse variation in more complicated relationships
 varies jointly means directly with other relationships
Examples of Joint Variation
Equation Form
y varies directly with the square of x.
y = kx2
y varies inversely with the cube of x.
k
y=
x3
z varies jointly with x and y.
z varies jointly with x and y and inversely with w.
z = kxy
kxy
z=
w
Example 4: z varies directly with x and inversely with y. When x = 12, y = 4 and z = 30, find z
when x = 6 and y = 8.
AII.9 Statistics:




Given data
enter the set of data into a list (or lists) on your graphing calculator.
Look at the scatterplot graph, decide which model is most reasonable (linear, quadratic,
cubic, logarithmic (LN), or exponential)
calculate the appropriate regression formula. STATCalc Write the equation of this
particular equation and use it to predict appropriate values not already included in the
data.
Example 1) {(1, 2.1), (3, 3.1), (5, 4.0), (7, 5.2), (9, 5.9)}
 Plug x into List 1 and y into List 2
 Graph the scatterplot.
 It should represent the line y  .485 x  1.635
Using the equation for the line of best fit, predict the y value when x = 6:
Plug 6 in for x.
y  .485(6)  1.635  4.545
Example 2) Based on this scatter plot, which regression model would be most appropriate? Write the
equation.
Example 3) What type of function would best model the data below? What is the equation for the
curve of best fit?
x
y
-2
-1
0
1
1
4
2
11
3
50
4
90
Equation: ____________________
AII.11 Normal Distribution:
A normal distribution shows data in a symmetrical, bell-shaped curve.
Data is centered around the mean (  )
The standard deviation (  ) tells how each data value in the set differs (deviates) from the mean.
Know from memory that the Empirical Rule tells us the probability distribution of the standard normal
curve.
____% of the data fall within one standard deviation of the mean.
____% of the data fall within two standard deviations of the mean.
____% of the data fall within three standard deviations of the mean.
Label the normal curve indicating the following: mean, deviation, and how much percent is in each
deviation.
Example 1. 300 Algebra II students took a statistics test. The scores were distributed normally with a
mean of 85 and a standard deviation of 5. Label the mean and three standard deviations from the mean
and answer the following questions.
a. What percentage of scores are
between 80 and 90?
b. What percentage of scores are
between 75 and 95?
c. How many students could be
expected to score between 60
and 75?
d. What percentage of scores is less than a score of 65?
e. What percentage of scores is greater than a score of 85?
Z-Score
A “z-score” represents the number of standard deviations away from the mean
 A z-score with a negative value lies below the mean.
 A z-score of 0 lies at the mean
 A z-score with a positive value lies above the mean.
Z-scores are a way to compare different normal distributions,
x
, where  is the mean and  is the standard deviation.
To calculate the value of a z-score, z 

“x” is the number you are seeking.
Reading/Applying the Z-Score Table:
Suppose I want to find out P( z  1.28) .
 Find the z-score of the data point (use the formula) if it is not given to you.
 Sketch and shade a graph of what data you are looking
 Go to the positive z-table since 1.28 is positive.
 Find 1.2 in the left row and
 Read across until you are under the .08 column
 What value did you find? _______
 This is the probability that z < 1.28.
The total amount under the normal curve is 1 (or 100%).
The mean is “0”


If the z-score is negative, use the negative table; if the table is positive use the positive table.
If you shaded to the left of the data point you are looking for take the value off the table.





If you shaded to the right of the data point, subtract the table number from 1 (1-table value)

If you are looking for a small portion of the data in the middle of the graph, take the far right zscore table value and subtract the far left z-score table value.
Example 2:
The mean height of eleventh-grade boys at Franklinton High School was 69.5 inches and the standard
deviation of the data was 3 inches.
The mean height of eleventh-grade girls at Franklinton High School was 64 inches and standard deviation
of 2.25 inches.
How many standard deviations away from the mean is a boy who is 65 inches and a girl who is 65 inches
tall?
Find the z-score of Boys:
Find the z-score of Girls:
z
z
65  69.5
 1.5
3
65  64
 0.44
2.25
Go to the z-table to get the corresponding % of data below the z-score found:
Solutions:
A boy that is 65 inches tall is below the mean by -1.5 deviations, or 0.0668 (6.68%) taller than his
classmates.
A girl who is 65 inches tall is above the mean by 0.44 deviation, or 0.6700 (67%) taller than her
classmates.
Example 4. The mean test score on a standardized test is 216; the standard deviation is 52. What
percentage of the scores are above 335?
# standard deviations from mean? ____
% = ____
Example 3. A group of 73 rabbits has a mean hopping distance of 67 in. and a standard deviation of 4 in.
How many frogs will hop more than 72 in.?
# standard deviations from mean? ____
Example 5:
P( z  1.28)
% = ____
Example 6:
P( z  1.87 )
# Rabits = ______
Example 7:
P( 2.32 z
1.17 )