Mar. 29 Statistic for the day: 80.4% of Penn State students drink; 55.2% engage in “high-

risk drinking” source: Pulse Survey, n = 1446, margin of error = 2.6%

Assignment:Assignment:Read Chapter 20Read Chapter 20

Exercises p265: 1, 2, 3a,bExercises p265: 1, 2, 3a,b

Sample means: measurement variables

Data from stat100.2 survey. Sample size 237.Mean value is 152.5 pounds.Standard deviation is about (240 – 100)/4 = 35

Suppose we want to estimate the mean weight at PSU

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Histogram of Weight, with Normal Curve

What is the uncertainty in the mean?

Suppose we take another sample of 237.

What will the mean be?

Will it be 152.5 again?

Probably not.

Consider what happens if we take 1000 sampleseach of size 237 and compute 1000 means.

Standard deviation is about (157 – 148)/4 = 9/4 = 2.25

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curve, based on samples of size 237Histogram of 1000 means with normal

Note: When we have measurement data and we considerthe sample mean, there are two different standard deviations:

1. The original standard deviation of the data. We estimatedthat from the original histogram of the data.

2. The standard deviation of the sample mean. We estimatedthat from a histogram of 1000 sample means.

In general we will have to be given the standard deviation of the data. Or we will have to estimate it from a histogram.

But once we have the standard deviation of the data (calledthe sample standard deviation) we can skip the histogram of sample means and use a formula.

Formula for estimating the standard deviation of the sample mean (don’t need histogram)

Suppose we have the standard deviation of the original sample. Then the standard deviationof the sample mean is:

standard deviation of the datasample size

Jargon: The standard deviation of the mean is also called the standard error or the standard error of the meanand abbreviated SEM or SE Mean.

So in our example of weights:

The standard deviation of the sample is about 35.Write SD = 35.

Sample size is 237

Hence by our formula: SEM = SD/square root of sample size

Standard error of the mean is 35 divided bythe square root of 237: SEM = 35/15.4 = 2.3

So the margin of error of the sample mean is 2x2.3 = 4.6

Report 152.5 + 4.6 or 147.9 to 157.1

147.9 152.5 157.1

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bell is centered at 152.5.The standard error is 2.3 and the Normal Curve of sample mean.

2 SEM

95% in middle

True pop mean in here someplacesample mean

Anatomy of a 95% conf idence interv al

Using the margin of error as 2 SEMs we really have a 95% confidence interval for the pop mean.

1. sample mean: 152.5 (given)

2. sample standard deviation: SD = 35 (given)

3. sample size: 237 (given)

4. standard error of the mean: SEM = 35/sqrt(237) = 2.3 (you calculate)

5. number of SEMs for 95% confidence coefficient: 2 (you look up in a normal z table)

Now you put it all together:6. 95% confidence interval for pop mean: 152.5 + 2x(2.3) 152.5 + 4.6

147.9 to 157.1

The steps for 95% confidence interval:

Example: Estimate the true population mean amountspent by stat 100 students for text books in fall 2001.Include a 98% confidence interval.

From the class sample survey

1. mean: 275 dollars2. sample standard deviation: SD = 120 dollars3. sample size: 100

4. standard error of the mean: SEM = SD/sqrt(100) = 120/10 = 12

5. number of SEMs for 98% confidence interval: 2.33

6. 98% confidence interval: 275 + 2.33x(12) 275 + 27.96 247.04 to 302.96

Interpretation: We estimate that the population of stat100 students spent about $275.

98% confidence interval is $247 to $303, a reasonable set of values for the pop mean.

So we believe that the true pop mean amount spenton books this semester is between $247 and $303 withour best guess of $275.

$303$275$247

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Normal Curve of sample mean.The standard error is $12 and thebell is centered at $275.

2.33 SEM

98% inmiddle

True pop mean in here someplacesample mean

Anatomy of a 98% confidenceinterval

Guess the next number in the sequence

1, 1, 2, 3, 5, 8,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …Called a Fibonacci sequence.

Ratios of pairs after a while equal approximately .618

eg. 8/13 = .61513/21 = .61921/34 = .61834/55 = .618

Fibonacci

width

length

618.lengthwidth

If

Then the rectangle is called the golden rectangle.

Daisy Head

21 clockwise spirals34 counterclockwise

Parthenon in Athens

Villa in Paris by Le Corbusier

St. Jerome

Leonardoda Vinci

La ParadeGeorges Seurat

Place de la ConcordePiet Mondrian

The golden rectangle has become an aesthetic standard for western civilization.

It appears in many places:architectureartpyramidsbusiness cardscredit cards

Research question: Do non-western cultures also incorporate the golden rectangle as an aesthetic standard?

Width to Length ratios for rectangles appearing on beaded baskets of the Shoshoni

0.693 0.662 0.690 0.606 0.570 0.749 0.652 0.628 0.609 0.844 0.654 0.615 0.668 0.601 0.576 0.670 0.606 0.611 0.553 0.633 0.625 0.610 0.600 0.633 0.595

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beaded basketsWidth to Length ratio of rectangles in Shoshoni

Golden Rectangle: .618

Question: Is the golden rectangle (.618) a reasonablevalue for the mean of the population of Shoshonirectangles?

1. sample mean: .6382. sample standard deviation: SD = .0613. sample size: 254. standard error of the mean: SEM = .012

(I calculated if for you.)

Could you create a 95% confidence interval for the population mean? (We’d like to know whether .618 is in this interval.)