Elementary Statistics

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Elementary Statistics

• Probability

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis (Chapter1)
• Collect data to test the theory-sampling method,
  and experimental design (Chapter 2, Chapter3)
• Analyze the results --- graphically, numerically
  (Chapter 4, 5)
• Use models Chapter 6
• Understand the language of probability Chapter 7
• Interpret the results and make a decision
  --p-value approach

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The language of Probability

• Sample Space all possible outcomes
• Event any subset of the sample space
• An event A is said to occur if any one of the
  outcomes in A occurs when the random process is
  performed once.
• Relation of event
• The union of two events
• The intersection of two events
• The complement of a event
• Two events A and B are disjoint or mutually
  exclusive if they have no outcomes in common.
  Thus, if one of the events occurs, the other
  cannot occur.

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Probability of an Event

• The probability of any event is the sum of the
  probabilities of the outcomes that make up that
  event.
• 0 ?P(A) ? 1
• If the outcomes in the sample space are equally
  likely to occur
• P(A) n(A)/n(S)

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Addition Rule
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Addition Rule
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Conditional Probability Chance under certain
condition

• Here is a random sample of 200 adults
• What is the probability that an adult selected at
  random has a college level of education given the
  adult is a female?
• Probability of A (college level) given the
  condition C(Female)

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Conditional Probability
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Independent

• If two events do not influence each other (that
  is, if knowing one has occurred does not change
  the probability of the other occurring), the
  events are independent.
• Two events A and B are independent if
• P(AB) P(A) or, equivalently, if P(BA)
  P(B).
• P(A and B) P(A)P(B) is A and B are independent.
• Are two events A and C in Education Example
  independent?
• Revisit Family Plan LDI 7.1, what should be the
  answer?

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Think about it

• Which of the following sequences of tosses is
  more
• likely to occur?
• (a) THTHHT (b) HHHTTT (c) HHHHHH
• Check your answer by finding the probability of
  each of the sequences.
• If we observe HHHHHH, is the next toss more
  likely to give a tail than a head?

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Mutually Exclusive and Independent Event

• A and B are exclusive A occurs than B doesnt
  occur.
• A and B are independent A and B do not
  influence each other.
• Example Tossing two coins
• A The first toss is head
• B The second toss is head
• C The first toss is tail

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis (Chapter1)
• Collect data to test the theory-sampling method,
  and experimental design (Chapter 2, Chapter3)
• Analyze the results --- graphically, numerically
  (Chapter 4, 5)
• Use models Chapter 6
• Understand the language of probability Chapter 7
• Interpret the results and make a decision
  --p-value approach
• Make a decision on proportion.- Chapter 8,9

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Parameter and Statistics

• What is the probability that adult selected at
  random is female? ---- 50
• Parameter
• What is the probability that adult selected at
  random from the sample of 200 adults is female?
  --- 112/200 56 (data from the Example 7.4)
• Statistics

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Sample Proportion --- Statistics

• Parameter --- Fixed
• Statistics --- various --- depend on the sample
• The proportion of woman -- variable
• Sample ASTA201 19/25
• Sample AMTH108 12/25
• Sample AMTH242 4/20
• Q which value should we trust?

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Big Picture
All kind of Samples With size n
Population p-- Parameter
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Sampling distribution of a statistics

• The sampling distribution of a statistic is the
  distribution of the values of the statistic in
  all possible samples of the same size n taken
  from the same population.

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P 50 pick a sample of 20

• Sample 1 M W M M M M W W M W M W M M M W W M W
  W
• Sample 2 W W M W M W W M M M M W M M W W W M W
  W
• Sample 3 M M W W W M M W M W M W M W W M M M W
  W

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Experiment --- LDI 8.1 P504

• Pick a random sample of four.
• Generate a random list of 0 and 1 , 0 man,
  1-woman
• Here is the list (Seed value 91)
• 1111 , 0101 , 1000, 1000, 1001, 0100, 1111,
  1010, 0110, 0010,1010, 0101,0010,0101,
  1101,1011,1110,0101,1011, 1111

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Summary
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(a) What was the most likely proportion of women
in the sample? 0.50 (b) What percentage
of the time did we get... ... 0
women, for a sample proportion of 0.00? 0
... 1 woman, for a sample
proportion of 0.25? 25 ... 2
women, for a sample proportion of 0.50? 35
... 3 women, for a sample proportion
of 0.75? 20 ... 4 women, for a
sample proportion of 1.00? 15 (c) What is
the sample distribution of the proportion? (
Histogram )
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Think about it

• A Larger Sample Size
• If we randomly select a sample of four people
  from a population with 50 women, it is quite
  likely to have one woman (25) in the sample, and
  it is possible to get all women in the sample.
  Suppose you increase the sample size to 20
  people.
• Would you be surprised if only 5 (25) of the 20
  selected individuals were women?
• Would you be surprised if all of the 20 selected
  individuals were women?

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Do you agree -- several facts about sample
proportion

• The large the sample size, the less the
  variability of sample proportion
• The center of the distribution of is the
  true parameter p for large sample
• The shape is approximately bell-shaped

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Q How tell us about p?

• Population USCA Students
• P is the proportion of male students (Unknown)
• Sample Convenience Samples
• All sections of ASTA201
• 4/24, 3/24, 4/24 , 5/24, 2/24
• All sections of AMTH221,222
• 0/24, 1/24, 2/24, 2/24
• What is the distribution of this statistics ?

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Bias and Variability

• A statistic is unbiased if the center of its
  sampling distribution is equal to the
  corresponding population parameter value.
• The variability of a statistic corresponds to the
  spread of its sampling distribution. A statistic
  whose distribution shows values that are very
  spread out and dispersed is said to lack
  precision.

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Lets Do It! 8.3 Three Estimators
The following histograms show the sampling
distributions of three estimators. The true
population parameter is 8.
Most of the time, we suppose our sample is
unbiased and representative
(a) Which estimator (s) is/are unbiased?
Circle your answer (s) I II III (b) Is
Estimator III more precise than Estimator I?
Circle one YES NO
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• What Do We Expect of Sample Proportions?
• The values of the sample proportion vary from
  random sample to random sample in a predictable
  way.
• When the sample size n is large, the sample
  proportion can take on many possible values in
  the range of 0 to 1, so the random variable can
  be viewed as a continuous random variable with a
  density curve as its model.
• When the sample size n is large, the distribution
  of can be modeled approximately with a normal
  distribution.
• The center of the distribution of the values is
  at the true proportion p (for any sample size n
  and any value of p).
• With a larger sample size n, the values tend to
  be closer to the true proportion p. That is, the
  values vary less around the true portion p. The
  variation also depends somewhat on the value of
  the true proportion p.

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Q How tell us about p?

• is approximated N(p, )
• If the sample size n is sufficiently large.
• np ? 5 and n(1-p) ? 5
• Large sample size -gt small variability
• Example Standard Deviation for
• N 24, P 0.50 ,
• N 100, P 0.50

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Example Proportion of voters in favor

• Suppose that of all the voters in a state, 30
  are in favor of Proposal A. Suppose a random
  sample of n 400 voters will be obtained, the
  proportion of sampled voters in favor of the
  proposal will be computed.
• Q What is the probability that this sample
  proportion is less than 30
• Q What is the probability that this sample
  proportion is less than 25
• Q What is the probability that this sample
  proportion is between 25 and 35

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9
12.5
N(0.09, 0.0143) 0.0072
No, the sample size is not large enough.