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Table A.1: Area to left for various values of Z

From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.Used with permission.

Try It! Scholastic Scores

a. What is probability a randomly selected school will score below 70?

Scholastic test given to all students in public schools grades 2-4 to produce an overall school score. Schools that score in top 20% are labeled excellent. Schools in bottom 25% are labeled "in danger" and schools in bottom 5% are designated as failing. Scores on test are approx normal with mean of 75 and standard deviation of 5.

Try It! Scholastic Scores

b. What is score cut-off required for schools to be labeled excellent?

Scores on test are approximately normal with mean of 75 and standard deviation of 5.

Note: Find percentile on standard normal scale = z, then convert it to corresponding percentile on X scale by: x = z+

8.7 Approximating Binomial Probabilities pg 61

Recall LH Problem (page 52) 10% of Americans LH

Let X = # of left-handed in a r.s. of 120 Americans.

X has Binomial distrib with n = 120 and p = 0.10 Mean = np = 120(0.10) = 12 Std. Dev. = = 3.29

Find probability the sample will contain 20 or fewer LH.

)90.0)(10.0(120)1( pnp

)20()19()1()0()20( XPXPXPXPXP

Normal Approximation to the Binomial

If X is a binomial random variable based on n trials with success probability p, and n is large, then the random variable X is also approximately …

Conditions:Approximation works well when both np and n(1 – p) are at least 10.

Try It! Returning to our Left-Handed Problem

Let X = # of left-handed in a r.s. of 120 Americans. X has Binomial distrib with n = 120 and p = 0.10 Mean = 12 and Std. Dev. = 3.29

a. Probability sample will contain 20 or fewer LH.

Since np = 120(0.10) = 12, n(1 – p) = 120(0.90) = 108 both are at least 10, can use normal approximation.

X has approximately a Normal distribution:

N( _____, ______ )

Try It! Returning to our Left-Handed Problem

a. Probability sample will contain 20 or fewer LH. X has approx a Normal distribution: N(12 , 3.29)

P(X ≤ 20) = ?

Try It! Returning to our Left-Handed Problem

b. How likely is it that more than 20% of the sample will be left-handed Americans?

Try this on your own – Chapter 8 notes go up on Ctools tonight!

Chapter 9: Introduction to Inference

9.1 Parameters, Statistics, and Inference pg 65

Some distinctions to keep in mind ... Population versus Sample Parameter versus Statistic

e.g. Population proportion versus sample proportion

e.g. Population mean versus sample mean

Statistical Inference: the use of sample data to make judgments or decisions about populations.

Let’s try a clicker question … Parameter or Statistic?

Q: A polling organization surveys 100 adults in Ann Arbor and finds 72% of those sampled favor tougher penalties for persons convicted of drunk driving.

A) Parameter

B) Statistic

The Big Five

9.2 From Curiosity to Questions About Parameters

From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.Used with permission.

Statistics as Random Variables pg 67

9.3 SD Module 0: Overview of Sampling Distributions

The value of a statistic from a random sample will vary from sample to sample.

So a statistic is a random variable and it will have a probability distribution.

This probability distribution is called the sampling

distribution of the statistic.

Definition: The distribution of all possible values of a statistic for repeated samples of the same size from a population is called the sampling distribution of the statistic.

Sampling Distribution for a Sample Proportion

Identify: Population =

Parameter =

Sample =

Statistic =

9.4 SD Module 1: Sampling Distribution for One Sample Proportion

Example: Poll conducted by Heldrich Center for Workforce Development (at Rutgers Univ). A probability sample of 1000 workers resulted in 460 (for 46%) stating they work more than 40 hours per week.

Sampling Distribution for a Sample Proportion

Can say how close this obs sample proportion is to the true population proportion p?

If we were to take another random sample of the same size n = 1000, would we get the same value for the sample proportion ?

What are possible values for if we took many random samples of the same size from this population? What would the distribution of the possible values look like?

What is the sampling distribution of ?

p̂

p̂

p̂

p̂

p̂

Aside: Can you Visualize it?

Our one = 460/1000 = 0.46 Take another r.s. get another value of _______ Repeat process over and over … Picture showing possible values n = _______

Observations:

p̂

p̂

What if n even larger? n = ____________

Our first = ____________ Repeat process over and over … Picture showing possible values n = _______

Observations:

p̂

Normal Approximation to the Binomial pg 68

Proportion = Count n

Recall: If X is a binomial random variable based on n trials with success probability p, and n is large, then the random variable X is also approximately …

Conditions:Approximation works well when both np and n(1 – p) are at least 10.

)1(, pnpnpN

nXp ˆ

Sampling Distribution of a Sample Proportion

If n is small, convert question to count and use binomial distribution.

If n is large, use a normal approximation.

Sampling Distribution of :

If the sample size n is large enough,

(namely both np and n(1 – p) are at least 10)

then the sample proportion is approximately …

n

pppN

)1(,

p̂

p̂

Try It! Do Americans really vote when say they do?

Telephone poll taken two days after election. 800 adults polled, 56% reported had voted. In press: only 39% of adults actually voted.

Suppose 39% rate is correct population proportion. Assume responses of 800 adults a random sample.

a. Describe/sketch Sampling Distribution of p̂

Try It! Do Americans really vote when say they do?

is approximately N(0.39, 0.017)

b. What is the approximate probability that a sample proportion who voted would be 56% or larger for a random sample of 800 adults?

p̂

c. Does it seem that the poll result of 56% simply reflects a sample that, by chance, voted with greater frequency than the general population?

A) Yes, just random variation.

B) No, something else may be going on.

C) I don’t know.