Sampling DistributionUsing the Z-table

Review: on using TI-84 Find the area of the following normal

distribution using the Z table

1. z > 2.85

2. z < 2.85

3. z > -1.66

4. -1.66< z <2.85

Syntax: 2nd>vars>normalcdf>min,max,0,1)

+infinity=1EE99-infinity=-1EE99

0.0022

0.9978

0.9515

0.9493

Review: on using TI-84 Find the area of the following normal

distribution using the Z table

Syntax: 2nd>vars>normalcdf>min,max,mean,sd)

+infinity=1EE99-infinity=-1EE99

0.2266

0.2260

0.6915

1. x > 0.40, mean: 0.37, sd: 0.042. 0.40<x< 0.50, mean: 0.37, sd:

0.043. 0.40<x< 0.50, mean: 0.41, sd:

0.02

Sampling distribution of sample proportion

p Count of success in sample

Size of the sample

X

n==

The mean of the sampling distribution is exactly p

p The standard deviation of the sampling

distribution is p

√p(1-p)n

One way of checking the effect of under-coverage, non response, and other sources of error in a sample survey is to compare the sample with the known facts about the population.

ƥ=0.11

ℳ=0.11

σ=0.00808

√p(1-p)nσ=

About 11% of American adults are black, the proportion of blacks in an SRS of 1500 adults should therefore be close to 0.11. It is unlikely to be exactly 0.11 because of sampling variability. If a national sample contains only 9.2% blacks should we suspect the sampling procedure is somehow under-representing blacks?

X=.092

probability

ƥ=0.092

P(ƥ ≤ 0.092)= 0.0129

Only 1.29% of all samples would have so few blacks. Therefore we have a good reason to suspect that the sampling procedure underrepresented blacks.

Rule of Thumb

1. You can only use the formula for the standard deviation of p-hat only when the population is at least 10 times as

large as the sampleN 10n≧

2. Use the normal approximation to the sampling distribution of p-hat for values of n and p that satisfy

np 10≧ and n(1-p) 10≧

Practice: Rule of thumb(s)

Explain why you cannot use the methods in ch9.1 on this problem

A factory employs 3000 unionized workers, of whom 30% are hispanic. The 15-member union executive committee contains 3 hispanics. What would be the probability of 3 or fewer Hispanics

if the executive committee were chosen at random from all the worker.

N 10n≧

3000 10(15)≧

3000 150≧satisfied

np 10≧ and n(1-p) 10≧

15(.30) 10 and 15(1-30) 10≧ ≧4.5 10 and 10.5 10≧ ≧

NOT

satisfied

Mean and Standard Deviation of a sample mean

Mean of sampling distribution: ℳx = ℳStandard Deviation of sampling

distribution: σx = σ / √n

Mean and Standard Deviation of a sample meanInvestors remember 1987 as the year the stocks lost 20% of their value in a single day. For 1987 as a whole, the mean return of all common stocks

on the NYSE was =-3.75% and the standard ℳdeviation of the returns was about σ=26%. What

are the mean and standard deviation of the distribution for all possible samples of 5 stocks?

ℳ= -3.75% or

-.00375

σx = σ / √n

σ= 26 / √5

σ= 11.6376%

Example: Servicing Air conditioners

The average of servicing an air conditioning unit in a certain company is 1 hour with a standard deviation of 1 hour as well. The company has been contracted to maintain 70 of these units in an apartment building. You must schedule technicians’ time for a visit to this building. Is it safe to budget an average of 1.1 hours for each unit? Or should you budget 1.25 hours?

ℳ=1, σ=1Sd=σ/√n

1/√70=0.120hrs

ℳ=1 hr. σ=.120 hr.

N(1, .120)

normalcdf(1.1 , +∞ , 1, .120)

P(x-bar>1.1 hrs.) P(x-bar>1.25 hrs.)

= 0.2033

normalcdf(1.25 , +∞ , 1, .120)

= 0.0182

If you budget 1.1 hrs, there is a 20% chance that the

technicians will not be able to complete the work

If you budget 1.25 hrs, there is a 2% chance that the technicians will not be able to complete the work

What you should have learned

1. Identify parameters and statistics in sample

experiment.

2. Recognize the fact of sampling variability.

3. Interpreting sampling distribution.

4. Describe the bias and variability of statistic in

terms of the mean and its spread.

5. Understand the variability of a statistic.

Statistic from larger samples are less variable.

A. Sampling Distribution

1. Recognize when a problem involves a sample

proportion ƥ

2. Find the mean and standard deviation of

sampling distribution

3. Know that as the spread gets smaller the

sample size gets bigger.

4. Recognize a reliable conclusion by verifying the

rule of thumbs: N≥10n and np≥10, nq≥10

B. Sample Proportions

1. Recognize when a problem involves the mean of

the sample. (x-bar)

2. Find the mean and sd of the sampling

distribution when the and σ of the population ℳare known.

3. Know that as the spread gets smaller the

sample size gets bigger

4. X-bar is approximately Normal when the

sample size is large (CLT)

C. Sample Means