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# Outlines - PowerPoint PPT Presentation

Stat 350 Lab Session GSI: Yizao Wang Section 016 Mon 2pm30-4pm MH 444-D Section 043 Wed 2pm30-4pm MH 444-B. Outlines. Binomial and normal distribution Sampling distribution and CLT (Module 4) Confidential intervals (Module 5, Actv.1) Permission to post forms

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### Stat 350 Lab SessionGSI: Yizao WangSection 016 Mon 2pm30-4pm MH 444-DSection 043 Wed 2pm30-4pm MH 444-B

Outlines
• Binomial and normal distribution
• Sampling distribution and CLT (Module 4)
• Confidential intervals (Module 5, Actv.1)
• Permission to post forms
• Today’s qwizdom questions are anonymous. You don’t have to login with your UMID.
Binomial Distribution

Example of B(n,p): coins flipping

Flip a coin n times. The probability of getting heads each time is p.

The number of heads we get during n times is a r.v. distributed as B(n,p)

Another classical example is giving a survey of one ‘yes/no’ question to n random selected persons.

Normal Approximation of Binomial Distribution

X ~ B(n,p)

• P(X = k) is decided by the parameters…but
• When n is large, very difficult to calculate!
• Approximation by normal distributionApproximately X ~ N( np,sqrt(np(1-p)) )
Normal Distribution
• Normal distribution is very rare in real world, but often a very good approximation, with some nice mathematical properties.
• Written as X ~ N(\mu,\sigma)
• Z-score (z-statistic) is the standardized X by Z = (X-\mu)//sigma
• Z ~ N(0,1) (why we want to standardize X?)
• What do the normal distributions look like? How to relate the shape with the two parameters?
Normal Distribution
• 10 minues In-lab review (8 questions) CTools\Lab Info\Lab review: Normal Distribution
Parameters vs. Statistics

Statistics are random variables. Parameters are constants.

Statistical Inference
• Population parameters are unknownconstants.
• Statistics are random variables obtained through sampling.
• Statistical inference: using statistics to estimateparameters.
• Statistics are also called estimators (of parameter).Example: X-bar is the estimator of μ
• We need to study the distribution of statistics.(Random variables have fixed distributions.)
Sampling Distribution
• The probability distribution of the sample statistics is called its sampling distribution.

The X in the pictures is not a random variable… Consider it as X-bar.

Statistical Inference

What kind of estimators do we prefer?

• Unbiased: the mean of estimator equals parameter.
• Small variation: small standard deviation.
Module 4
• Objectives: study the influence of the sample size and the distribution of parent population on the sampling distribution.
• Sampling Distribution Applet (CTools/lab info)
Summary
• The shape of the sampling distribution will depend on the distribution of original parent population as well as the sample size.
• The sampling distribution is approximatelynormal when…
4(a) Sampling Dist. of the Sample Mean

If the parent popul. is a normal dist. with a mean μ and a stand. dev. σ, then for anysample size, the sample mean will have a

__________ dist. with a mean of _____

and a stand. dev. of _____.

4(b) Central Limit Theorem

If the parent popul. is NOTa normal dist. but with a mean μ and a stand. dev. σ, then for a largesample size, the sample mean will have a

__________ dist. with a mean of _____

and a stand. dev. of _____.

• Shape of parent popul.
• Shape of dist. of sample mean
• Standard deviation of sample mean
• Sample size
True or False
• If n is large, the sample data will always have a normal distribution.

Confidence Interval

Recall the parameter-statistic comparison…

• We never know the true population parameter value.
• We use a one-sample (with several observations) statistic to estimate it.
• A sample statistic may not be exactly equal to the corresponding parameter value. (why confidence interval?)
Confidence Interval

Example: we are 95% confident that the true parameter value lies inside the confidence interval [a, b].

Confidence interval provides a method of stating:

• What interval tells:How close the value of a statistic is likely to be to the value of a parameter
• What confidence tells:The accuracy of it being that close
Confidence Interval

Basic structure for any confidence interval:

estimate multiplier standard error

Margin of error. The

Bigger the margin of

error, the wider the CI (why?)

The sample statistics

such as p-hat, x-bar.

Confidence Interval

Two interpretations:

• A 95% Confidence Interval: We are 95% confident that the true parameter value lies inside the confidence interval. The interval provides a range of reasonable values for the population parameter.
• The 95% Confidence Level: If the procedure were repeated many times (that is, if we repeatedly took a random sample of the same size and computed the 95% confidence interval for each sample), we would expect 95% of the resulting confidence intervals to contain the true population parameter.
Confidence Interval

Principles for using CIs to guide decision making:

• Principle 1: A value notin a CI can be rejected as possible value of the population parameter. A value in a CI is an “acceptable” or “reasonable” possibility for the value of a population parameter.
• Principle 2: When the CIs for parameters for two different populations do not overlap, it is reasonable to conclude that the parameters for the two populations are different.
Confidence Interval
• The probability that the true parameter lies in a particular, already computed, confidence interval is either 0 or 1. The interval is now fixed and the parameter is not random, so the parameter is either in that particular interval or it is not.
Module 5 Activity1
• Good summary on p26
• Confidence Interval for Mean Applet (CTools/Lab Info)
• We are 95% confident that the popul. mean will be in the computed confidence interval.
• The computed confidence interval will contain the popul. mean 95% of the time.
• 95% of all confidence intervals created with this method are expected to contain the popul. mean.
Before we finish today…