Mr. Rogers - AP Statistics Objectives
Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
11 t-Test 12 Inf for Prop 13 Chi Test 14 Regression HT  
Unit Plan Practice Test
Latin

Latin/Greek Root Words

arch--------->ancient, example: archtype;         chrono------>time, example: chronology;             -dom----------->quantity/state, example: freedom               fer-------->carry, example: transfer;               gen--------->birth, example: generate;                 luc-------->light, example lucid;                 neo--------->new, example: neonatologist;                olig--------->few, example: oligarchy;              omni--------->all, omniscient;            sym--------->together, symbol;

(Statistics connection)

AP Statistics Standards

IV. Statistical Inference: Confirming models and testing hypotheses (30% –40%)

B. Tests of significance

C. Special case of normally distributed data

  1.  t-distribution
  2. Single sample t procedures
  3. Two sample (independent and matched pairs) t procedures

 

Objectives

 

Essential Question: How does the US justice system compare to statistical analysis?

  Hypothesis or Significance Testing

  1. State the question asked by a significance test and the two possible answers. Is there clear evidence of an effect?

  2. State a generic null hypothesis.
  3. There is no clear evidence of an effect.

    There is nothing new or out of the ordinary.

    The status quo exists.

  4. State a generic alternative hypothesis.
  5. There is clear evidence of an effect.

  6. Give the null and alternative hypothesis for the American justice system.

    Null Hypothesis: not guilty. This is the status quo for most people.

    Alternative Hypothesis: guilty

  7. Define P-value.

Assuming Ho is true, the probability of obtaining a test statistic as extreme or more extreme than the one obtained is ______.

The smaller the p-value, the stronger the evidence is against Ho

  1. State the statistic used for indicating the level of significance. (Hint: it begins with a "p".)

  2. State the type of distribution used for tests of significance. The sampling distribution.

  3. Describe one tail and two tail tests from the standpoint of the null hypothesis and the p-values.

  4. Be as one with "z-test for a population mean".

  5. Perform "z-test for a population mean" in the following ways:

  • by hand (using the calculator only for basic mathematics) using z-tables.

  • by hand, using the calculator only for basic mathematics and finding areas

  • using the hypothesis testing features of the TI-83

  • using Minitab software

Homefun (formative/summative assessment) Exercise 1, 5, 7, 9, 15, 17 pp. 546 to 547

 
Essential Question: Can a statistically significant hypothesis have no practical value?

Alpha Levels

  1. Define the significance level, alpha--predetermined maximum acceptable p-value for rejecting the null hypothesis.

  2. Use one and two tailed tests of significance.

one-tailed: area of tail = alpha

two-tailed: area of each tail = (alpha) / 2

 

  1. Use alpha to evaluate statistical significance.

  2. Use a confidence interval (confidence level = C) as a significance or hypothesis test.

    significance level = (1 - C)

    a confidence interval is essentially identical to a 2-tail hypothesis test.

    Formative Assessment: What two types of statistical tools are used for inference and what type of distribution is used for making inferences?

 

  1. Describe the difference between statistical and practical significance. Even a tiny difference between x-bar of a sample and a population will be statistically significant if the sample size is large enough. Such a tiny difference may have no practical significance.

  2. State when statistical inference is not valid. (When based on data from a poorly designed study or experiment.)

Homefun (formative/summative assessment)
Essential Question: How many ways can you make an error of judgement?

Types of Errors

  1. Pass the ultimate test of true statistics nerdhood: Explain the difference between type 1 and type 2 errors.
  2. Explain what a type 1 and type 2 error is for the American justice system.
Ho: innocent, Ha: guilty
type I error: punish an innocent person
type 2 error: let a guilty person go free
  1. Explain what a type 1 and type 2 error is for quality control in manufacturing.
Ho: the product is acceptable to the customer
Ha: the product is unacceptable to the customer
type I error: reject acceptable product and don't ship it.
type 2 error: ship unacceptable product to the customer
  1. Generate a truth table for a hypothesis test.

  2. State how alpha relates to the type 1 error. Whether a one or two tailed test, alpha always = type 1 error.

  3. Name the hypothesis which is considered true when determining the probability of having a type 1 error. Ho

  4. Name the hypothesis which is considered true when determining the probability of having having a type 2 error. Ha

  5. Identify the areas representing the probabilities of type 2 and type 1 errors on a diagram of a hypothesis test showing a hypothetical sampling distribution.

  6. Determine alpha and beta. beta = type 2 error

Power given: β = (1 - power)

 

Special Cases for β = Type 2 Error

μHa = μHo: β  = (1 - α)

μHa = (α boundary): β  = 50%

 

  1. Determine the power of a hypothesis test. power = (1 - β)

Beta given: power = (1 - β)

 

Special Cases for Power

μHa = μHo: power = α

μHa = (α boundary): power = 50%

 

  1. Plot and interpret a power curve (power vs. separation between μHo and μHa) for a hypothesis test.

  • sigmoidal shape

  • at zero separation power =  alpha

  • asymptote at 100% (as separation approaches infinity, power approaches 100%)

  1. State how power can be applied to quality testing in manufacturing.

 

Homefun (formative/summative assessment) Exercise 21, 23, 25, 27, 29 pp. 548 to 549

Summative Assessment: Test Objectives 1-27

 

Essential Question: How does a confidence interval for proportions compare to one for means?

Ch. 12.1 Inference for Proportions

  1. State the meaning of p-hat. A statistic estimating a population proportion

p-hat =

   count of successes in sample 
 count of observations in sample
  1. Calculate the mean and standard deviation of a binomial distribution.

Data Type Mean Std Dev
count or number np [np(1 - p)]^0.5
proportion p [p(1 - p) / n]^0.5
  1. Be aware that a binomial distribution (the distribution typically used for analyzing proportions) is essentially a sampling distribution. Note that as a sampling distribution, when the sample size is large enough (see below), the distribution begins to resemble a normal distribution.

  2. When appropriate, correctly model a binomial distribution as a normal distribution if the 2 conditions shown below are met.

np 10

n(1-p) 10

 
  1. Perform a hypothesis test comparing a single large sample proportion (p-hat) against a know population proportion (p). Note, this is a one proportion z-test.

Z =   (p-hat) - p
[p(1-p)/n]0.05

 

Homefun (formative/summative assessment)Exercise 35, 39, 41, 43, 47, 57, 59 pp. 562 to 565

 

Essential Question: How can we perform significance tests with small samples and an unknown standard deviation for the population?

Estimating a Population Mean When Its Standard Deviation is Unknown

  1. State the 2 assumptions that need to be met for using a t-test.

  • SRS

  • Normal Distribution of the population (symmetrical with single peak)

  1. Calculate standard error of a t-statistic

SE = s / ( n^.5 ) .

  1. If the sample size is large, the z-interval can be used with the above SE.

  2. Explain when a t statistic is used rather than a z-score.
  • Population Standard deviation not known
  • Sample Size is Small
  1. Calculate t statistics.

  2. State the degrees of freedom for a one-sample t-test. df = ( n-1 )

 
  1. Perform one sample t-procedures:

  • by hand (using the calculator only for basic mathematics) using t-tables.

  • by hand, using the calculator only for basic mathematics and finding areas.

tcdf (L,U,D)
Lower t-value
Upper t- value
Degrees of Freedom
  • using the hypothesis testing features of the TI-83. Note that

μ is the mean associated with Ha

μo is the mean associated with Ho

  • using Minitab software
  1. Use the t-distribution with the following limitations:
Sample Size
Skew
Nearly N-Distr
Outliers
Less than 15
None
Yes
No
At least 15
Minor
Yes, minor skew ok
No
At least 30
Significant
Yes, skew ok
No

 

Homefun (formative/summative assessment):

 

 

Essential Question: What is the single most powerful form of hypothesis testing and why?

Matched Pairs Testing (Spanish Camp)

  1. Apply the t-test of significance to matched pairs (MP) situations.
    • MP tests are 1 sample t-tests

    • Ho: μ = 0

    • df = (number of pairs - 1)

    • Be aware: you must establish that the population is approximately normally distributed.

How to Establish That a Population is Normally Distributed

Box Plot - symmetrical, box width smaller than a whisker's width

Normal Quantile Plot - straight line

  1. Explain what is meant by a robust test or confidence interval. p-value  or confidence interval changes little if assumptions are violated.

  2. Be as one with the information in the "using t procedures" box on p..

  3. Be aware that outliers are very harmful to the t-test. Making a modified box plot of the data in order to look for outliers is a very good idea.

 

 

Homefun (formative/summative assessment): Exercise 65, 67, 73, 75, 77 pp. 587 to 589; Work the entire AP Statistics Practice Test pp. 597 to 599

 

Essential Question: How can you determine if 2 populations differ if at the start you have no information?

Two-Sample t-Test

  1. State the assumptions made for two-sample tests. (p. 650)

    SRS used for generating the sample

    independent - matched pairs violate independence

    normally distributed population

  2. Create confidence intervals and hypothesis test using two sample t procedures assuming that the sigmas of the two populations are unequal. This is the most conservative assumption.

Ho: μ1 - μ2 = 0  but can also be written, Ho: μ1 = μ2

 
Test Statistic for  Hypothesis Testing Margin of Error for Confidence Interval
t = (xbar1 - xbar2) - 0
(s12/n1 + s22/n2)0.5
ME = t* (s12/n1 + s22/n2) 0.5

  1. Create confidence intervals and hypothesis test using two sample t procedures and the most conservative method of determining df.

df = (the lower of n1 -1 or n2 - 1)

  1. Be aware of the more accurate way to calculate df as is done by the TI-83 calculator. This method can return a df that includes a decimal fraction. For example: df = 12.37

  2. Perform two sample hypothesis t-procedures on the TI-83.

  3. State the key assumption required for using the pooled two-sample t-procedures. This is the method is an option in the TI-83 calculator.

The sigmas of the two populations are the same

 

Homefun (formative/summative assessment): Exercise

 
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