Normal Distributions 2/27/12

transcript

• Normal Distribution• Central Limit Theorem• Normal distributions for confidence intervals• Normal distributions for p-values• Standard Normal

Corresponding Sections: 5.1, 5.2

Exam 1 Grades

slope (thousandths)-60 -40 -20 0 20 40 60

Measures from Scrambled RestaurantTips Dot Plot

r-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Measures from Scrambled Collection 1 Dot Plot

Nullxbar98.2 98.3 98.4 98.5 98.6 98.7 98.8 98.9 99.0

Measures from Sample of BodyTemp50 Dot Plot

Diff-4 -3 -2 -1 0 1 2 3 4

Measures from Scrambled CaffeineTaps Dot Plot

xbar26 27 28 29 30 31 32

Measures from Sample of CommuteAtlanta Dot Plot

Slope :Restaurant tips

Correlation: Malevolent uniforms

Mean :Body Temperatures

Diff means: Finger taps

Mean : Atlanta commutes

phat0.3 0.4 0.5 0.6 0.7 0.8

Measures from Sample of Collection 1 Dot PlotProportion : Owners/dogs

What do you notice?

All bell-shaped distributions!

Bootstrap and Randomization Distributions

• The symmetric, bell-shaped curve we have seen for almost all of our bootstrap and randomization distributions is called a normal distribution

Normal Distribution

-3 -2 -1 0 1 2 3

Central Limit Theorem!

For a sufficiently large sample size, the distribution of sample

statistics for a mean or a proportion is normal

http://onlinestatbook.com/stat_sim/sampling_dist/index.html

Central Limit Theorem

• The central limit theorem holds for ANY original distribution, although “sufficiently large sample size” varies

• The more skewed the original distribution is (the farther from normal), the larger the sample size has to be for the CLT to work

Central Limit Theorem

• For distributions of a quantitative variable that are not very skewed and without large outliers, n ≥ 30 is usually sufficient to use the CLT

• For distributions of a categorical variable, counts of at least 10 within each category is usually sufficient to use the CLT

• The normal distribution is fully characterized by it’s mean and standard deviation

Normal Distribution

mean,standard deviationN

Normal Distribution

0.523,0.048N

Bootstrap DistributionsIf a bootstrap distribution is approximately normally distributed, we can write it as

a) N(parameter, sd)b) N(statistic, sd)c) N(parameter, se)d) N(statistic, se)sd = standard deviation of variablese = standard error = standard deviation of statistic

Confidence Intervals

If the bootstrap distribution is normal:

To find a P% confidence interval , we just need to find the middle P% of the distribution

N(statistic, SE)

Best Picture

What proportion of visitors to www.naplesnews.com thought The Artist should win best picture?

ˆ .15p

Best Picture

www.lock5stat.com/statkey

Area under a Curve• The area under the curve of a normal distribution is equal to the proportion of the distribution falling within that range

• Knowing just the mean and standard deviation of a normal distribution allows you to calculate areas in the tails and percentiles

http://davidmlane.com/hyperstat/z_table.html

Best Picturehttp://davidmlane.com/hyperstat/z_table.html

Best Picture

For a normal sampling distribution, we can also use the formula

to give a 95% confidence interval.

sample statistic 2 SE

0.096.15

For normal bootstrap distributions, the formula

gives a 95% confidence interval.

How would you use the N(0,1) normal distribution to find the appropriate multiplier for other levels of confidence?

sample statistic 2 SE

For a P% confidence interval, use

where P% of a N(0,1) distribution is between –z* and z*

*sample statistic z SE

Find z* for a 99% confidence interval.

z* = 2.576

News Sources“A new national survey shows that the majority (64%) of American adults use at least three different types of media every week to get news and information about their local community”

The standard error for this statistic is 1%

Find a 99% confidence interval for the true proportion.Source: http://pewresearch.org/databank/dailynumber/?NumberID=1331

News Sources*sample statistic z SE

2.5760.64 0 0. 1

0 0.6 64 .02

0.614,0.666

Confidence Interval Formula

*sample statistic z SE

From original data

From bootstrap

distribution

From N(0,1)

First Born Children

• Are first born children actually smarter?

• Based on data from last semester’s class survey, we’ll test whether first born children score significantly higher on the SAT

• From a randomization distribution, we find SE = 37

first born not first born 30.26X X

First Born Children

What normal distribution should we use to find the p-value?

a) N(30.26, 37)b) N(37, 30.26)c) N(0, 37)d) N(0, 30.26)

first born not first born 30.26, 37SX X E

Hypothesis TestingDistribution of Statistic Assuming Null

Statistic

-3 -2 -1 0 1 2 3

Observed Statistic

Distribution of Statistic Assuming Null

Statistic

-3 -2 -1 0 1 2 3

Statistic

-3 -2 -1 0 1 2 3

Observed Statistic

p-value

p-valuesIf the randomization distribution is normal:

To calculate a p-value, we just need to find the area in the appropriate tail(s) beyond the observed statistic of the distribution

N(null value, SE)

First Born ChildrenN(0, 37)

p-value = 0.207

First Born Children

Standard Normal• Sometimes, it is easier to just use one normal distribution to do inference

• The standard normal distribution is the normal distribution with mean 0 and standard deviation 1

Statistic

-3 -2 -1 0 1 2 3

Standardized Test Statistic

• The standardized test statistic is the number of standard errors a statistic is from the null value

• The standardized test statistic (also called a z-statistic) is compared to N(0,1)

sample statistic null valueSE

p-value

1) Find the standardized test statistic:

2) The p-value is the area in the tail(s) beyond z for a standard normal distribution

First Born Children

1) Find the standardized test statistic

30.26 037

First Born Children

2) Find the area in the tail(s) beyond z for a standard normal distribution

p-value = 0.207

z-statistic

• Calculating the number of standard errors a statistic is from the null value allows us to assess extremity on a common scale

Formula for p-values

From randomization

distribution

From H0

From original data

Compare z to N(0,1) for p-value

Standard Error

• Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations?

• We can!!!

• Or rather, we’ll be able to on Wednesday!

Normal Distributions 2/27/12

Documents