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The Normal Distribution

n = 20,290  = 2622.0  = 2037.9 Population

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

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

1000 samples Sampling distribution of the mean

Non-normal Approximately normal

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

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

Distribution of means of samples with n =10

Larger samples equal smaller standard errors

Central limit theorem

Button pushing times Frequency Time (ms)

Distribution of means

Binomial Distribution

Normal approximation to the binomial distribution

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.

Inference about means Because is normally distributed:

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!

t has a Student’s t distribution }

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

Confidence interval for a mean

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

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)

Estimate the mean and standard deviation

Find the standard error

Table A3.3

Find the critical value of t

Putting it all together...

99% confidence interval

Confidence interval for the variance

Table A3.1

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

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

One-sample t-test

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.

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

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

Human body temperature

Degrees of freedom df = n-1 = 23

Comparing t to its distribution to find the P-value

A portion of the t table

-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.

Body temperature revisited: n = 130

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.

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

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