AP Stat Ch 2

Question 0

Strategy for exploring data:

Answer 0

Always plot data: make a graph (histogram or stemplot)
Look for overall pattern (shape, center, spread) and for outliers
Calculate a numerical summary to briefly describe center and spread
Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. The curve is a mathematical model for the distribution. It is an idealized description that gives an overall pattern of the data but ignores minor irregularities as well as any outliers.

Question 1

Empirical Rule

Answer 1

In a normal distribution, 68% of the data is within 1 standard deviation of the mean, 95% is within 2 standard deviations and 99.7% is within 3 standard deviations.

Question 2

How value of standard deviation affects bell curve

Answer 2

Larger the standard deviation, the wider the curve is.

Bell curve with s=1 is much taller and narrower than one with S=3

Question 3

Percentile

Answer 3

One way to describe performance, or location in a distribution, is to use percentiles.

The pth percentile of a distribution is the value with p percent of the observations less than it.

Question 4

Example of percentile:

Jenny’s score is the 22nd highest score. There are 25 scores. What percentile did she score.

Answer 4

There are 21 values less than Jenny’s– 21/25 = 84th percentile.

N-1 / total

Question 5

Another example of percentile:
Katie got the highest score in the class on the exam. There are 25 people in the class.

Answer 5

24 scores less than Katie.

24/25= 96th percentile.

Question 6

Relative cumulative frequency

Answer 6

Instead of wanting to know which percent of the data falls into a particular class, we often want to know which percent falls below a certain value. To make this possible, we will compute the relative cumulative frequency for each class, which is the sum of the relative frequency of that class and all the classes below it.

Add up relative frequency from groups at or below–this sum is the percent that falls at or above a value

Question 7

Example of relative cumulative frequency:
Say that 2 presidents were inaugurated from 40-44, 7 from 45-49, and 13 from 50-54, …
44 total presidents
What’s the RCF in the 40-44, 45-49, 50-54 groups?

Answer 7

In 40-44, 2/44 presidents = 4.5%. So rel f = 4.5%. RCF = 4.5% too
45-49– 7/44 presidents, rel f =15.9%. RCF = 15.9+4.5 = 20.5%. 20.5% of the presidents inaugurated were 49 or less
50-54 – 13/44 = 29.5%. RCF = 20.5+29.5=50%. 50% of the presidents were 54 or less when inaugurated.

Question 8

Ogive

Answer 8

Graph of a cumulative relative frequency distribution is referred to as an ogive.

Question 9

How to graph an ogive

Answer 9

Label and scale your axes and title your graph
Plot a point corresponding to the RCF in each class interval at the left endpoint of the next class interval. For example, plot a point at 4.5% above the age value 45 to indicate that 4.5% of presidents were inaugurated before they were 45 years old. 
Begging your ogive with an height of 0% at the left endpoint of the lowest class interval. 
Last point should be at a height of 100%
Y axis is RCF and x axes is the variable-- for presidents it is the ages
4.5% of presidents in 40-44. So point at (45,4.5%)

Question 10

Standardizing

Answer 10

Another way to describe position is to tell how many standard deviations above or below the mean it is. Converting scores from original values to standard deviation units is known as standardizing.

Question 11

Why do we standardize?

Answer 11

To allow for us to compare different approx. normal distributions.
Every set of data has different set of values. For example, heights of people might range from 18 inches to 8 feet and weights can range from one pound to 500 pounds. Those wide ranges make it difficult to analyze data so we standardize the normal curve, setting it to have a mean of zero and a standard deviation of one.

Question 12

Standardized score/ z score

Answer 12

Z score tells us how many standard deviations an observation is from the mean, and in which direction.

Question 13

How to calculate z score

Answer 13

(X- x bar)/ s

Value - mean divided by standard deviation.

Question 14

Example of z score:
Jenny got an 86 on stat test. Mean is 80 and S = 6.07.
82 on physics test. Mean is 76, s = 4.
Which did she performs better relative to her class?

Answer 14

ZSTAT = 86-80 / 6.07 = .99
ZPHYS = 82-76 / 4 = 1.5
Higher z score on physics, so did better relative to her class on the physics.

Question 15

How does standardization affect the distribution?

Answer 15

When we subtract the mean of the data from every data value, the mean is now zero
When we divide each of these shifted values by s, the standard deviation should be divided by s as well. Since the standard deviation was S to start with the new standard deviation becomes 1.

Shape the same
Center - mean now is zero
Spread - S is now one

Question 16

Density curve

Answer 16

A curve that is always on or above the x axis and has an area of exactly one underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that interval.
Density curves, like distributions, come in many shapes. Normal, skewed left, or skewed right. Outliers are not described by density curves. Curve is an approx that is easy to use and accurate enough for practical use.

Question 17

Advantage of density curve over histogram

Answer 17

Doesn’t depend on our choice of classes

Question 18

Mean and median of a density curve

Answer 18

Mean is the balance point– hard to do this by eye– calculate mathematical way and locate on curve.
Median–equal areas point– divides area in half.

Median and mean are the same for symmetric density curve. Center of the curve. Mean of a skewed curve is pulled away from the median in the direction of the long tail.

Question 19

When to use x bar and S vs sigma and mu

Answer 19

Use English letters for statistics– measures on a data set.
Greek letter for parameters – idealized distribution.

Question 20

Normal distributions

Answer 20

Density curves that are symmetric, single peaked, and bell shaped are called normal curves, and they describe normal distributions.
A normal distribution curve shows the frequency (how many times something occurs) in a symmetric graph.
Following properties:
Graph is the highest at the mean
Mean = Median = mode
Data is symmetrical about the mean

Question 21

Inflection points

Answer 21

Points on a curve at which there’s a change in curvature, and are one standard deviation away from the mean.

Question 22

What type of data is normally not normal

Answer 22

Income. Skewed right

Question 23

Example using empirical rule:
A battery has an average life span of 50 hours, S=3. Normally distributed
A. What percent of batteries last at least 44 hours?
B. If we have 1500 batteries, how many are within one standard deviation of the mean

Answer 23

A. 44 hours is -2s.
95% of data within two S of mean– 47.5% between -2s and mu and 50% greater than mu. So 97.5%
B. 68% within one S of the mean – .68*1500 = 1020 batteries.

Question 24

Standard normal model

Answer 24

The normal distribution N(0,1) with mean 0 and standard deviation 1 is called the standard normal model. Remember that standardizing won’t change shape– if not normal before, standardizing won’t change shape.

Question 25

Shorthand for normal dist.

Answer 25

N (mean, standard deviation)

Question 26

Standard normal z table

Answer 26

Table of areas under the standard normal curve. Tells us the area under the curve to the left of z. Also the percentile of z.

Question 27

How to use the z table

Answer 27

First convert data to z scores
Then looking at a portion of the z table above, if the z score is -2.15, we find the z score by looking down the left column for the first two digitis, -2.1, and then across top row for third digit, 5. Table gives the percentile .0158. That means 1.58% of the z scores are less than -2.15.

Question 28

Example of analyzing z scores:
If at student scored a 720 on SAT math and mean is 500 and standard deviation is 100, what is his z score and what’s it mean according to the z table.

Answer 28

Z = (720-500)/100=2.2

This means that the score is 2.2 standard deviations ABOVE the mean

Question 29

Example of z table:

What z score represents the first quartiles in a normal model? Find the cut point for the 25th percentile.

Answer 29

Look at z table – look for .25 – Q1 –> z= -.67. Then use the z formula to get the corresponding score in the distribution.

Question 30

Normal probability plot

Answer 30

Used to assess normality. If it is linear, then the data set is roughly Normal.