Review for Test 3 Math 1231

1
Review for Test 3Math 1231
This test covers the following chapters Ch 18-23
2
Random Variables and Sampling Distributions

• Sample means are used to estimate population
  means.
• Law of Large Numbers Averaged observed outcome
  (sample mean) guaranteed to be close to
  population mean in the LONG RUN (large sample
  size/many repetitions)
• Statistics vary in repeated samples.
• The distribution of sample proportions is the
  sampling distribution- describes values taken on
  by RV in all possible samples.
• Statistics from larger samples are less variable!

3
Sampling Distributions

• The sampling distribution model shows the
  behavior of the sample proportion over ALL the
  possible samples.
• As sample size increases, the sampling
  distribution becomes LESS VARIABLE (ie, sigma
  decreases!).
• The Central Limit Theorem (CLT) tells us that
  the sampling distribution of the sample
  proportion (or mean) is approximately Normal for
  large n, regardless of the population, as long as
  observations are independent.

4
Sampling Distribution Proportions

• The sampling distribution of the sample
  proportion has
• Normal distribution, N(p, )
• Mean equal to the population proportion, p
• Standard deviation equal to
• z-score
• The statistic that estimate the true proportion
  of something is the sample proportion, phat.
• phat count of successes/n
• Assumptions/Conditions Random Sample
• Independence of values
• Sample less than 10 pop.

5
More Sampling Distributions Means

• The sampling distribution of sample mean has
• Mean equal to the population mean!
• Standard Deviation equal to population SD /
  sqrt(n)
• Example Population is N(10,16)
• The sampling dist of the mean of sample of size 4
  has mean 10 and SD 16/sqrt(4) 8.
• The larger the sample, the smaller the SD of the
  sample mean, at rate of sqrt(n) (Central Limit
  Theorem)

6
Finding Probabilities with Sampling Distributions

• Can use Normal Distribution to calculate
  probabilities.
• Chapter 6 If Population is N(10,16), then
  P(xlt12) is found using
• Look up z 0.125 in z-table, about 0.55. This is
  probability of xlt12!
• Chapter 18 Same Population, N(10,16). Sample of
  size 4 this time.
• Find P(xbarlt12). This is not the same as P(xlt12).
• Look up z0.25 in z-table, 0.5987. This is Prob.
  of xbarlt12.
• Other methods from Ch6 work the same way, plus Ch
  14 rules!
• P(xbargt12)1-P(xbarlt12)

7
Confidence Intervals Whats the Big Idea?

• Confidence Intervals- estimate the unknown value
  of a POPULATION parameter
• Most general form estimate - margin of error
• The statistic that estimates the true proportion
  of something for a population is the sample
  proportion, phat.
• phat count of successes/n
• The CI for proportions takes the form
• phat - z SE(phat)
• Where Standard Error, SE(phat)
• Given specified ME, find sample size needed
• Remember to round n up- you generally cant
  sample partial things!

8
More on CIs

• How do I get z?
• You have a confidence level, C, that has been
  specified or chosen by you.
• This means we want to capture the middle C
  proportion of the area under the density curve.
• How much of curve not caught? 100-C
• Half of that amount is in each tail.
• Use the Inverse Normal Tool, enter the area in
  the tail.
• Or, Look INSIDE the Z table for the proportion
  closest to that area. Then see what z corresponds
  to that area.

9
Misc on CIs

• A 95 CI means that we are 95 confident that our
  interval has captured the true mean of the
  population, mu.
• A Lower confidence level results in a smaller
  margin of error (smaller z- less area under
  curve captured)
• A Larger sample results in a smaller margin of
  error, less variability in sampling distribution
• Assumptions
• Sample must be Random
• Sample must be independent.
• Sample lt 10 of population
• At least 10 successes and 10 failures (nphat and
  nqhat gt10)

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Proportion example

• Ex Tossed coin 10,000 times. Got 5067 heads.
• Phat5067/10000 0.5067.
• Give 95 CI for true proportion z1.96 for 95
  confidence.
• CI
• We are 95 confident that true proportion of
  heads is
• 0.4969 lt p lt 0.5165

11
Hypothesis Tests for Proportions

• Hypothesis Tests- assess evidence about a claim
  about a population do we think a claim is true
  or not?

12
Hypothesis Tests

• Null Hypothesis Ho Statement of NO
  Difference/no effect
• p
• Alternative Hypothesis the alternate to Ho, Ha.
• Can be one-sided or two-sided.
• One-sided Ha p gt
• p lt
• Two-sided Ha p ¹
• P-value How likely are we to have picked a
  sample with a proportion as far from the null
  hypothesis as we did, or further away, under the
  assumption of Ho being true? This is the area
  under the curve for the calculated z.

13
Process for Hypothesis Test
Step 1) Identify the parameter you are
testing. Step 2) Identify what type of test you
need to run 1 sample one proportion z-test
Step 3) State null and alternative
hypotheses. Ho p Ha choices p lt
pgt or p ¹ Step 4) Check Assumptions Random
Sample? Independent values? sample lt 10
population? npgt 10 and nqgt10?
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Process continued

• Step 5) Calculate test statistic, z
• z(phat p)/(s(phat))
• Step 6) Sketch P-value and find from Z-table
• Remember that a two-sided Ha doubles the P-value
  you find from the table.
• Step 7) Is the P-value lower than your alpha
  level? (is it Statistically significant?)
• If P-value lt alpha Reject Ho
• If P-value gt alpha Fail to reject Ho.
• Know how to write conclusion in real words.

15
Proportion example

• Ex Tossed coin 10,000 times. Got 5067 heads.
• Phat5067/10000 0.5067.
• Hyp Test Was the coin fair? Ho p0.5 (coin
  fair) Ha p¹0.5 (not fair)
• Calculate z
• Find P-value (this is 2-sided)
• From Table, P(zlt1.34) 0.9099
• P-value 2 (1-0.9099) 0.1802
• Our P-value is gt alpha (0.05), so we fail to
  reject the null.
• Yes the coin is fair!

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Errors

• Type I error You reject Ho when it is actually
  true.
• Probability of Type I error same as alpha level
• Sending an innocent person to jail.
• Type II error You fail to reject Ho when it is
  actually false.
• Letting a guilty person go free.

17
CIs and HTs for Means

• Since we dont typically know the SD of the
  population, we must use s, the SD of the sample,
  in our calculations.
• t (xbar mu)/ (s/ sqrt(n))
• Distribution of t is NOT Normal
• SE(xbar)s/sqrt(n)
• CI for mean
• Xbar - t s/sqrt(n)

18
Finding t

• Use t-table. Locate confidence level at bottom.
• df n-1 (n is sample size)
• Go up to row for your df- that value is t
• Once you have t, simply plug in to the
  expression for the CI to get the confidence
  interval!
• Everything else works the same as CIs for
  proportions!
• Additional condition if population dist is not
  known to be Normal, then n must be gt 40, or
  histogram of the sample should be symmetric and
  unimodal.

19
Process for Hypothesis Test
Step 1) Identify the parameter you are
testing. Step 2) Identify what type of test you
need to run 1 sample t-test (dont know
sigma) 2 samples (ch 24-25, for Final) -are
they paired up/linked? Paired test Mu(1-2)
-samples not linked or
different sizes- 2-sample test
mu1-mu2 Step 3) State null and alternative
hypotheses. Ho mu Ha choices mu lt
mugt or mu ¹
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Process continued

• Step 4) Calculate test statistic, t.
• t(xbar mu)/(s/sqrt(n))
• Step 5) Find P-value from T Table- you wont get
  an approximate value, but a range, usually
• Remember that a two-sided Ha doubles the P-value
  you find from the table.
• Step 6) Is the P-value lower than your alpha
  level? (is it Statistically significant?)
• If P-value lt alpha Reject Ho
• If P-value gt alpha Fail to reject Ho.
• Know how to write conclusion in real words.

21
Hyp. Test Example
The residents in a neighborhood would like to
know if the mean speeds on their road exceed the
posted speed limit of 30 mph. They sample 23
cars, and obtain a sample mean of 31.0 mph and a
standard deviation of 4.25 mph. 1) Want to test
the true mean speed of cars on the road. 2) Type
test? (one-sample) t-test of individual mus 3)
State Ho and Ha Ho mu 30 mph Ha mu gt 30 mph
4) Calc. t t 1.128 5) Find P-value df
22 Looking on T-table P-value is between 0.10
and 0.15 (if you use a calculator, youll get
0.136) 6) At a significance level of 0.05, we
fail to reject Ho. There is not enough evidence
to say the average speed is too high.
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Study Suggestions

• If you havent already, DO THE HOMEWORK!
• Review Quizzes and MathXL Homework, particularly
  any questions you missed.
• Solutions to all quizzes are online
• Think about what the big idea is that each
  question is trying to ask.
• Work Practice Test from Website
• Work Suggested problems in the Reviews in the
  book
• Email or IM Dr. Matos with questions, or use
  office hours!
• Make sure youve gone through ActivStats!