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The Normal Distribution

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n = 20,290 = 2622.0 = 2037.9 Population

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Y = 2675.4 s = 1539.2 Y = 2588.8 s = 1620.5 Y = 2702.4 s = 1727.1 Y = 2767.2 s = 2044.7 SAMPLES

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Y = 2675.4 s = 1539.2 Y = 2588.8 s = 1620.5 Y = 2702.4 s = 1727.1 Y = 2767.2 s = 2044.7 Sampling distribution of the mean

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1000 samples Sampling distribution of the mean

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Non-normal Approximately normal

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Sample means are normally distributed The mean of the sample means is . The standard deviation of the sample means is *If the variable itself is normally distributed, or sample size (n) is large

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Standard error The standard error of an estimate of a mean is the standard deviation of the distribution of sample means We can approximate this by

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Distribution of means of samples with n =10

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Larger samples equal smaller standard errors

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Central limit theorem

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Button pushing times Frequency Time (ms)

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Distribution of means

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Binomial Distribution

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Normal approximation to the binomial distribution

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Example A scientist wants to determine if a loonie is a fair coin. He carries out an experiment where he flips the coin 1,000,000 times, and counts the number of heads. Heads come up 543,123 times. Using these data, test the fairness of the loonie.

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Inference about means Because is normally distributed:

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But... We don’t know A good approximation to the standard normal is then: Because we estimated s, t is not exactly a standard normal!

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t has a Student’s t distribution }

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Degrees of freedom df = n - 1 Degrees of freedom for the student’s t distribution for a sample mean:

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Confidence interval for a mean

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(2) = 2-tailed significance level Df = degrees of freedom SE Y = standard error of the mean

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95% confidence interval for a mean Example: Paradise flying snakes 0.9, 1.4, 1.2, 1.2, 1.3, 2.0, 1.4, 1.6 Undulation rates (in Hz)

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Estimate the mean and standard deviation

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Find the standard error

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Table A3.3

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Find the critical value of t

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Putting it all together...

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99% confidence interval

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Confidence interval for the variance

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Table A3.1

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95% confidence interval for the variance of flying snake undulation rate

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95% confidence interval for the standard deviation of flying snake undulation rate

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One-sample t-test

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Hypotheses for one-sample t-tests H 0 : The mean of the population is 0. H A : The mean of the population is not 0.

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Test statistic for one-sample t-test 0 is the mean value proposed by H 0

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Example: Human body temperature H 0 : Mean healthy human body temperature is 98.6ºF H A : Mean healthy human body temperature is not 98.6ºF

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Human body temperature

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Degrees of freedom df = n-1 = 23

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Comparing t to its distribution to find the P-value

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A portion of the t table

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-1.67 is closer to 0 than -2.07, so P > With these data, we cannot reject the null hypothesis that the mean human body temperature is 98.6.

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Body temperature revisited: n = 130

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t is further out in the tail than the critical value, so we could reject the null hypothesis. Human body temperature is not 98.6ºF.

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One-sample t-test: Assumptions The variable is normally distributed. The sample is a random sample.

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