AP Statistics Review Notes

Unit 1

Discrete variables: Usually come from a count; it's a countable number of values.
Continuous variables: Usually come from a measurement.
Resistant statistics: Not greatly affected by outliers (e.g., median, IQR).
Non-resistant statistics: Influenced by outliers (e.g., mean, standard deviation, range).
Shape: Skewed, Symmetric, Bimodal, Uniform
Outliers: Discuss if obvious.
Center: Mean/median
Spread: Range, IQR, Standard Deviation (include context).
Positive skew: Mean > median
Negative skew: Mean < median
Frequency: How many observations in a category (e.g., 142 people answered "yes").
Relative frequency: The proportion of observations (e.g., 142 out of 250 answered yes).
Use mean for center when distribution is approximately normal. Use median if distribution is skewed or has outliers.
Outlier boundaries: An observation is an outlier if it lies outside the range of $[Q1 - 1.5(IQR), Q3 + 1.5(IQR)]$ .
Z-score formula: $z = \frac{x - \mu}{\sigma}$
Empirical Rule (for approximately normal distributions):
- 68% of observations are within 1 standard deviation of the mean.
- 95% are within 2 standard deviations.
- 99.7% are within 3 standard deviations.
Multiplying every value of the data set by a constant "C" stretches/compresses the data.
- Measures of position and spread change (x => xC, Mean => MC, St. Dev => St. Dev*|C|).
Five-number summary: Min, Q1, Median, Q3, Max
$\bar{x}$ = mean of a sample (statistic)
$s$ = standard deviation of a sample. Formula: $s = \sqrt{\frac{\sum(x - \bar{x})^2}{n-1}}$ .
$\mu$ = mean of a population (parameter)
$\sigma$ = standard deviation of population. Formula: $\sigma = \sqrt{\frac{\sum(x - \mu)^2}{N}}$ .
Adding a constant to every value in a data set changes measures of position, but not spread.
Finding who did better with respect to their groups:
1. Find each z-score.
2. Figure out where "better" is on a curve.
3. Choose z-score furthest from the mean in the direction of "better."
Interpretation:
- Mean: $\mu$ units is a typical value of the variable.
- Standard Deviation: $\sigma$ units is the typical amount an observed value of the variable deviates from the mean of $\mu$ units.
- Z-score: The number of standard deviations above/below the mean.
- IQR accounts for ~50% of the data.

Unit 6

Standard Error (SE): The estimate for standard deviation used when the population parameter is unknown (on formula sheet).
Interpretation of Confidence Level: Ex: In repeated random samples of size 45, approximately 99% of confidence intervals created will capture the true proportion of students who would be late to class.
Power = 1 - P(Type II error). To increase power, decrease P(Type II error).
Power increases when:
- n increases.
- Standard Deviation/SE decreases.
- $\alpha$ increases.
Margin of Error: How much a value of a sample statistic is likely to vary from the value of the population parameter (in formula sheet).
As sample size increases, the margin of error decreases, and the confidence interval becomes narrower.
To find the minimum sample size needed to get a specific ME:
- ME: If $\hat{p}$ is unknown, use $\hat{p} = 0.5$ .
Interpretation of P-value: "The probability of getting results like ours or more extreme is (P-value), assuming H0 is true."
$Pc = \frac{x1 + x2}{n1 + n2}$ , $Pc$ is needed for 2-sample z-test for a population proportion.
As confidence increases, margin of error increases, and the confidence interval gets wider.
One-sided tests use < or > in Ha; two-sided tests use ≠ in Ha.
P(Type I error) = $\alpha$
P.A.N.T.C. (for confidence intervals)
P.H.A.N.T.O.M.S. (for hypothesis tests)
If you aren't sampling, you don't need to check the 10% rule.
Finding Critical Value using InvNorm: $C = 95\% \implies \alpha = 0.05 \implies \alpha/2 = 0.025$ . InvNorm(0.025, 0, 1) = -1.96, so z = 1.96.
Finding n (Sample Size):

Confidence Interval Example

A student randomly sampled 182 other students at a large university to determine if the majority were in favor of building a new field house. 75 students were in favor. Make a 95% confidence interval of the true proportion.
$\hat{p}$ = true proportion of all students in favor of field house.
1. Random sample (stated).
2. 182 < 0.1(all students) ✓ reasonable.
3. $n\hat{p} = 182 * 0.412 = 75 > 10$ , $n(1-\hat{p}) = 182 * 0.588 > 10$ Sampling distribution is approximately normal.
One sample z interval for population proportion.
$0.412 \pm 1.96 \sqrt{\frac{0.412 * 0.588}{182}} \implies (0.340, 0.484)$
- (C = 95% use bottom line of t table >= 1.96)
I am 95% confident the true proportion of all students in favor of the field house is between 0.34 and 0.484.
We estimate that $\hat{p}$ = 0.75 in a previous study. How big of a sample do we need to be 95% confident we will be within 0.05 of our true population proportion?
- ME = 0.05, C = 95% (z = 1.96)
- $ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
- $0.05 = 1.96 \sqrt{\frac{0.75 * 0.25}{n}} \implies n \approx 288.12 \implies n = 289$

Unit 5

Point estimate: estimate for parameter (Ex. $\hat{s}$ estimates $\sigma$ , $\bar{x}$ estimates $\mu$ , $\hat{p}$ estimates p)
$\mu{\bar{x}} = \mu$ , $\sigma{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ . As n increases, $\sigma_{\bar{x}}$ decreases.
$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ .
CLT: n >= 30 (applies to sample means only).
As n increases, shape tends to get more normal.
The mean of a sampling distribution is not affected by sample size.
$np \geq 10$ and $n(1-p) > 10$ means sampling distribution is approximately normal.
1. Random 2. 10% (n < 0.1N) 3. Approx. normal ( $np \geq 10$ )
The probability that something is an exact number (Ex. 2 or 4) is 0 by definition (There is no area under a line).
The z-score is how many standard deviations above or below the mean something is.
"Probability" is the same as area.
Most likely will use Inverse norm when you have a percentile, mean, and Standard Deviation.
A normal z-score = N(0, 1). Look for UB and LB in every problem to use the Cdf functions.
An unbiased estimator is equal to the value of the parameter it's estimating.

Normal Curves Practice Problems

If a normal distribution has a mean of 50 and a standard deviation of 14, what percentile is X = 40?
- N(50, 14). normcdf(-999, 40, 50, 14) = 0.2375
- $z = \frac{40-50}{14} = -0.71$ .
- Approximately 24th percentile.
Find the probability that a random participant is taller than 71 inches. (mean: 68.4, Standard Deviation: 2.9)
- P(height > 71) = 0.1849
- N(68.4, 2.9). normcdf(71, 999, 68.4, 2.9) = 0.1849
You take a Random Sample of 20 people. What is the probability that 4 of them will be taller than 71 inches?
- n = 20, p = 0.1849
- Binomial: x = # of people over 71" (of 20)
- P(x = 4) = BinomialPdf(20, 0.1849, 4) = 0.2149
George got a 750 on the test and was in the 96th percentile. The mean score was 600. What was the Standard Deviation?
- x = 750, $\mu$ = 600, $\sigma$ = ?
- Find z-score @96th percentile
- InvNorm(0.96, 0, 1) = 1.75
- $z = \frac{x - \mu}{\sigma} \implies 1.75 = \frac{750 - 600}{\sigma} \implies \sigma = 85.7$

Unit 4

Finding Mean and Standard Deviation of a Probability Distribution:
- Outcome: Death (10000), Disability (5000), Neither (0)
- Probability: 0.001, 0.003, 0.996
- $\mu_x = 10000(0.001) + 5000(0.003) + 0(0.996) = 25$ (formula on sheet)
- $\sigma_x = \sqrt{(10000-25)^2(0.001) + (5000-25)^2(0.003) + (0-25)^2(0.996)} = 36.78$
Both formulas are on the formula sheet.
TI-Nspire: To put all that in your Calculator?
1. Put the x's and p's into lists.
2. Stat - Stat calc! - 1 var stats:
  - X, list = x
  - Free list = 0
  - list = p
Number of lists = 1
To find variance: $(σ{y+x})^2 = (σy)^2 + (σ_x)^2$
Binomials have 'n'. Geometric does not.
Geometric: If I roll a die until I get an even #, what's the prob that the first ever occurs on the 4th roll?
- P(1st even on 4th) = P(x = 4) = (odd)(odd)(odd)(even) = 0.5^3 * 0.5 = 0.0625
60% of people wear Green on St. Patrick's day. What is the prob you see someone wearing green before the 3rd person you encounter?
- P(green before 3rd) = P(1st Green) + P(2nd green)
- = 0.6 + (0.4)(0.6) = 0.84
Geometric and Binomial only work in independent trials.
What is the prob that at least 4 of the next flights will be overbooked?
- P(x>=4) = 0.332 + 0.114 + 0.044 + 0.007 + … = 0.497. Use binomial cdf to calculate the cumulative result and subtract from 1.

Setting up a Simulation

If Johnny usually hits 72% of his free throw shots, what is the prob he will hit 8 out of his next 10 shots?
1. Let the integers 0-71 = hit and 72-94 = miss.
2. Use RNG to generate 10 integers from 0-99. Repeats are ok.
3. Count how many of the 10 integers are between 0-71 and record the result.
4. Repeat process many times. Count the # of times you get "8 out of 10" and divide by the # of trials to estimate prob.
Probability without replacement: A Deck has 52 cards. Suppose you draw 2 cards without replacement. what is the prob that both cards are hearts?
- (13/52)(12/51) = 0.059
Tree diagram: 60% of AHS teachers are females and 45% of those teachers have a master's degree; of the male teachers, 55% have moster's
- Female: Master's (0.45), Don't (0.55)
- Male: Master's (0.55), Don't (0.45)

	Master's	Don't Have	Total
Female	0.27	0.33	0.6
Male	0.22	0.18	0.4
Total	0.49	0.51

Are the events "Prefers dogs" and "Children" independent?

	Dogs	Cats	No Pref.	Total
Children	36	8	2	46
No Pref.	22	26	18	54
Total	58	34	20	100

If independent, expect P(dogs) = P(dogs | children).
- Actually: P(days) = 58/100 = 0.58
- P(days | ch.) = 36/46 ≈ 0.78
You spin the Spinner 4 times
1. what is the probability of Getting Blue each time?
  - $BBBB: (0.25)^4 = 0.004$
2. what is the probability of not getting blue until the 4th spin?
  - $BBB'B = (0.75)^3 * (0.25) = 0.105$
3. Probability of getting at least 1 blue ? P(no blue) = (0.75)^4 = 0.316. P(at least blue) = 1 - 0.316 = 0.684

U4 Review Notes

Mutually Exclusive (disjoint): Events A and B have no shared outcomes, P(A ∩ B) = 0.
Joint probability (intersection) Formulas: Probability that A and B will both occur (overlap). P(A ∩ B)
- If ind. : P(A)= P(A|B). $P(A\cap B) = P(A) * P(B)$
Empirical probability: Relative frequency of something that happened. (Ex: 100 coin flips had 54heads 54/100).
TI-Nspire Geometric PDF vs. CDF:
- PDF: "For success on xth trial," inputs = p = -, X = - (menu-Statistics - Distributions)
- CDF: "For range" inputs: p:-, LB = -, UB=
Probability distribution is for a numerical outcome (describe using SOCS.).
$P(A\cap B) = P(A) * P(B|A)$
P(at least 1) = 1 - P(none)
Independent: If 2 events are independent, we know $P(A) = P(A|B), P(B) = P(B|A)$ , and $P(A\cap B)= P(A) * P(B)$ .
A Binomial deals with P(success or failure).
Conditional Probability: $P(A|B) = \frac{P(A\cap B)}{P(B)}$
complement: $P(E^c) = 1-P(E)$
TI-Nspire: Geometric Probabilities:
- Exactly 5 x=5 Geometric Pdf
- At most 7: X<=7 Geometric Cdf
- At least 7: X>=7 Cdf
- Less than 4 x<4 Cdf
- More than 4 x>4 Cdf
To combine random variables: You would add every possibility of combos. Ex.) X=[1,2,3] Y=[10,12]. X+Y= {11, 12, 13, 13, 14, 15 }
P(neither A nor B) = 1 - P(A or B)
Same concept for binomials (Except X = # of successes out of n trails).
Sample Space: Set of all possible non-overlapping outcomes (Ex. Process: Flip a coin 2x, Sample space: {HH, HT, TH, TT})
Cumulative prob.: Prob. of being less than or equal to each value of the random variable.
Ex.) P(roll a 6) = 1/6=0.167. If we rolled the dice many, many times, we would expect 0.167 of the rolls to be a six.
Binomial: BINS conditions
- B: Binary? (success or fail)
- I: Independent?
- N: # of trials?
- S: Success! (Prob. of success is the same for each trial).
Geometric: BITS
- B: Binary? (success or fail)
- I: Independent
- T: Trials (# of trials with success)
- S: Success (P(success) is the same for each trial).
Simulated Probabilities tend to get closer to the true prob. as the # of trials increases.

U3 Practice Problems

Bias identification: A mayer of a large city wants to know if more citizens prefer a Sports complex or a music complex. The mayer sends interns to stand outside the local YMCA and interview every 5th person (Systematic) who enters by asking which complex they would prefer. Why is this bad sampling!
Location: YMCA is a sports facility, so people tend to say "sports" more often then "music" than the general population. % "sports" is probably higher than truth.

U3 Review Notes

Sampling Techniques
- Systematic random sample: "Every kth". 1. # each individual 1 to N. 2. Choose a random starting place. 3. select every kth individual to include in the sample. (Fixed periodic interval).
- Cluster random sample: "Select all individuals in a few groups." 1. divide population into clusters that are similar in composition. 2. Take a SRS to select entire clusters. 3. use all individuals in each selected cluster in the sample.
- Stratified random sample: "select a few from each group." 1. divide population into groups based on a shared attribute,. 2. Take SRS within each group. 3. Combine the selected individuals from each group to make the sample.
- Simple Random Sample (SRS): # each individual and use a random technique to select.
- Convienience Sample: "Hey you!" No true random /probability based technique is used. Individuals are easy to reach or contact.
Experiment Designs
- Randomized block: Experimental units are grouped into "blocks" based on a shared attribute. All treatments are randomly assigned to individuals within each block.
- Matched pairs (pairs of individuals): Subjects are paired with a similar individual. Each pair receives both treatments. (1 individual gets Treatment 1 and the other gets Treatment 2.) (Ex. twins)
- Matched pairs (experimental unit acts as own pair): Subject receives both treatments.
Types of Bias
- Response bias: Bias resulting from problems in data instrument of process (Ex. Confusing questions, interviewer influence, lies, or miscalibrated measuring tools).
- Undercoverage bias: Part of a population has a reduced chance of being included in the sample.
- Non-response bias: individuals are selected, but refuse to respond or data can't be collected.
- Voluntary response bias: Sample is made up only of volunteers. (Ex. too many people who feel very strongly about an issue.)
In a SRS, every GROUP of size n has an equal chance of being chosen. (Every Group, not Individual).
If treatments are Randomly assigned to individuals, Causation conclusions are valid.
Random sampling Samples can be generalized to population.
OBSERVATIONAL Studies will never cause anything (no cause / effect).
Cluster Samples are better than SRS, be they are chosen from groups that already exist (neighborhoods, schools, stores, etc). They are faster and more convenient than SRSs.
Statistically Significant means the results were unlikely to have occurred by chance.
Well design Experiment: 1. Comparison of at least 2 treatment groups. 2. Random assignment 3 Replication (more than 1 experimental unit per treatment.) 4 Control of confounding variables.
Stratified samples are better than SRS, be they insure individuals From each strata are represented.

U2 Practice Problems

Height: 74, 65, 77, 72, 68
Weight: 256, 124, 320, 185, 158
r = 0.961: There is a strong positive linear relationship between Height and Weight.
ln(ŷ) = 7 - 0.045x, where x = fuel efficiency. Find ŷ when x = 20
- ln(y) = 7 - 0.045(20) = 6.1
- ŷ = e^6.1 = 445.84
Temp(degrees F): 57, 75, 43, 68, 67
#sold: 17, 12, 20, 8, 13
The mean temp. is 62°F, and mean #sold is 14.4. what is the correlation between temp, and # sold?
1. Put table into X and Y lists.
2. Menu - Stat - stat. calc. - 2-var-stats
3. Fill in x and y " lists and enter
4. Scroll to find "r": -0.85
ŷ = -3.4 + 5.2x and r = 0.66. What is the closest proportion of the variation in y that cannot be explained by the explanatory variable?
- $(1-r^2) \implies (1 - 0.66^2) \implies 1 - 0.4356 \implies 0.5644 \implies 56\%$ .

U1 Practice Problems

The distribution of fuel economy is unimodal and skewed left with outliers at 16mpg and 40mpg. The median is 31mpg and range is 24mpg.
- Shape? unimodal. Outliers? (16 and 40). Center? (median). Spread? (range)
Home prices (in thousands):
- 100, 110, 120, 130, 130, 150,
- 160 170 170, 190, 210, 230,
- 235, 240, 245, 260, 270, 270,
- Stem Plot:
  - 10 | 01233
  - 15 | 6779
  - 20 | 13344
  - 25 | 677899
    - Key: 10 = 100 thousand
Skewed distributions are best described by median, while Symmetric distributions are best described by mean.
Variance = Standard Deviation Squared = $\Sigma\frac{(x-\bar{x})}{n-1}$
The scores of an SAT test are a bell shaped distribution with a man of 1100 and a standard deviation of 200.
1. what is the z-score for a test score of 1300?
 - $z = \frac{1300 - 1100}{200} = 1$
2. What was the Z-score of a Score of 750?
 - $z = \frac{750 - 1100}{200} = \frac{-350}{200} = -1.75$
 - P(z< -1.75) = .04
- Finding probabilities. P(z>1.22) = 11.12%
- P(z1-2)=97.73%
- P(1.2<z<2.25) = 10.18
A reasonable for heights of randomly selected 5-year olds in normally distributed with a mean of 100cm and a standard deviation of Gam, What is the z-score for Ana, a 5-year old with a height of 94cm? What does this mean?
- $z = \frac{94 - 100}{6} = -1$
- Ana's height is 1 Standard Deviation below the mean height of 5-year-olds.

Unit 2 Review

If a residual plot has no pattern, a linear model is a good fit for the data.
Linear Regression: a+bx.
1. Enter x and y data in a spreadsheet
2. Menu-stats- Stat calculations - 4: Lin Reg a+bx Gives: a, b, r, and r^2 (coefficient of determination.
Extrapolation: Using LSRL to make predictions outside the domain of x.
TI-Nspire 1 variable stats:
- 1. Enter x and y data in a spreadsheet. 2. Menu - Stats - Stat. Calculations - 1 variable stats
- Gives: a, b, r, and mean and standard Deviation. for x and y.
Finding Correlation (r): Use b to determine sign (-/+)
Constant Coef SEcoef T P
height -10.94 S = Standard Deviation
r^2 = 90.3%
r = -√.903 = -0.95
r: between -1 and 1
close to 0 doesn't mean "no relationship…" could have Strong non-linear relationship.
Interpretation: There is a [strength] [Direction] linear relationship between x and y. Strong: |r|>=.8 . weak: |r|<.5 or moderate.
If a residual plot has Fanning, a linear model is appropriate, BUT we should proceed with caution.
residual = y - ŷ
Interpretation. % of variation in Y can be explained by x.
Correlation DOES NOT always imply Causation.
Association: Variables are associated if the dist. if different for each category of both variables. (compair using %'s as evidence)
High leverage points just have an unusual x-value as compared to the rest of the data.