7_confdnce_inter1

7_confdnce_inter1 - Last few slides from last time Example 3 What is the probability that p will fall in a certain range given p Flip a coin 50

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Last few slides from last time…
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Example 3: What is the probability that p’ will fall in a certain range, given p? Flip a coin 50 times. If the coin is fair (p=0.5), what is the probability of getting an estimate, p’, greater than or equal to 0.7 (=35 heads). E(P’) = 0.5 Std. error(P’) = sqrt((.5)(.5)/50) = .0707 z = (0.7-0.5)/0.0707 2.83 P(z > 2.83) (1-0.9953)/2 = 0.0024 = P(p’ > 0.7) Pretty unlikely to get such high estimates of p
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More examples on finding the mean and standard deviation of a r.v. •x ~ N ( µ x , σ x ), y~ N ( µ y , σ y ) • Z = x + 4y + 2 – E(Z) = E(x) + E(4y) + E(2) = µ x + 4 µ y + 2 – Var(Z) = var(x) + var(4y) + var(2) 2 = σ x 2 + 16 σ y •Z = ( 2 x 1 + 2x 2 -y)/5 – E(Z) = (E(2x) + E(2x) - E(y))/5 = 4/5 µ x –1/5 µ y – Var(Z) = var(2x/5) + var(2x/5) + var(y/5) = 8/25 σ x 2 + σ y 2 /25
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Confidence intervals 9.07 2/26/2004
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From last time… • Sampling theory – Start with a large population . – Take a sample of N units from the population, and compute a statistic on that sample. • E.G. statistic = sample mean = estimate of the population mean. – We imagine doing this many times, and make deductions about how the sample statistic (estimator) is distributed.
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The purpose of confidence intervals To estimate an unknown population parameter with an indication of how accurate the estimate is and of how confident we are that our estimate correctly captures the true value of the parameter.
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Relationship of confidence intervals to sampling theory • Confidence intervals – Start with one sample of N units from a population. – We make inferences from this one sample about the parameters of the underlying population .
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Deduction: Reasoning from a hypothesis to a conclusion. Induction: Reasoning from a set of observations to a reasonable hypothesis. Deduce properties of the sample statistic from knowledge of the population. sample population Infer properties of the population parameter from knowledge of the sample.
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Probability that the statistic takes on a particular range of values. sample population Why confidence vs. probability? The population parameter has a true value, which either is or is not in the given range. Confidence that the population parameter falls within a particular range of values.
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Archery & confidence intervals Suppose you have an archer who can hit the 10 cm radius bull’s eye 95% of the time. One arrow in 20 misses. 10cm
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Archery & confidence intervals You look at the target from the back. The archer shoots a single arrow. Knowing the archer’s ability, what can you say about where you expect the center of the bull’s eye to be?
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Archery & confidence intervals Well, 95% of the time, that arrow is going to be within 10cm of the center of the bull’s eye. Thus, 95% of the time, the center of the bull’s eye is going to be within 10cm of the arrow.
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Archery & confidence intervals Draw a 10cm circle centered around the arrow.
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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7_confdnce_inter1 - Last few slides from last time Example 3 What is the probability that p will fall in a certain range given p Flip a coin 50

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