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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

  • 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
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
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
  • 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 distribution1
Normal Distribution
  • 10 minues In-lab review (8 questions) CTools\Lab Info\Lab review: Normal Distribution
parameters vs statistics
Parameters vs. Statistics

Statistics are random variables. Parameters are constants.

statistical inference
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
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 inference1
Statistical Inference

What kind of estimators do we prefer?

  • Unbiased: the mean of estimator equals parameter.
  • Small variation: small standard deviation.
module 4
Module 4
  • Task 1-3
  • Objectives: study the influence of the sample size and the distribution of parent population on the sampling distribution.
  • Sampling Distribution Applet (CTools/lab info)
  • 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
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
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 _____.

what is the distinction between 4 a and 4 b choose all that apply
What is the distinction between 4(a) and 4(b)?Choose all that apply...
  • Shape of parent popul.
  • Shape of dist. of sample mean
  • Standard deviation of sample mean
  • Sample size
true or false
True or False
  • If n is large, the sample data will always have a normal distribution.

Clicker in your answer.

confidence interval
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 interval1
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 interval2
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 interval3
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 interval4
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 interval5
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
Module 5 Activity1
  • Good summary on p26
  • Confidence Interval for Mean Applet (CTools/Lab Info)
4 interpret the 95 confidence level in terms of a popul mean
# 4: Interpret the (95%) confidence level in terms of a popul. mean.
  • 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
Before we finish today…

Questions or comments?