Sampling Distribution, Chp. 9

1. Know the difference between a parameter and a statistic.

Parameter - A parameter is a number that describes the population. In most all cases the actual parameter in not known b/c the entire population cannot be examined.Proportion = Mean =

Statistic – A statistic is a number that is calculated from the sample data to estimate an unknown parameter.Proportion = Mean =

p

p̂ x

The ball bearing in a large container have mean diameter 2.5003 centimeters. This is within the specifications for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have mean diameter 2.5009 cm. Because this is outside the specified limits, the container is mistakenly rejected.

The Bureau of Labor Statistics last month interviewed 60,000 members of the U.S. labor force of whom 7.2% were unemployed.

Sampling Distribution –The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

• Unbiased Statistic / Unbiased Estimator.A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

• Variability of a Statistic.The variability of a statistic is the spread of its sampling distribution. This spread is determined by the sampling design and the size of the sample. The larger the sample the smaller the spread. As long as the population is much larger than the sample ( 10 times larger) the spread of the sampling distribution should be approximately the same for any population size.

• What type of probability question could be asked?

From a population, 60% approve of gambling, determine the probability that:a. You will get a sample of 70% or more that

approve of gambling with a sample size of 25.b. You will get a sample of 70% or more that

approve of gambling with a sample size of 100.

Penny Activity

• 1. This activity begins by plotting the distribution of ages (in years) of pennies. Sketch a density curve that you and your partner think will capture the shape of the distribution of ages of the pennies.

• 0 1 2 3 4 5 6 7 8 9 ………… (age of pennies)

Year Age Frequency

2014 0

2013 1

2012 (and so on) 2

• 3. Put your 25 pennies in a cup, and randomly select 5 pennies. Find the average age of the 5 pennies in your sample, and record the mean age as (5). 2 times

• 4. Repeat step 3, except this time randomly select 10 pennies. Calculate the average age of the sample of 10 pennies, and record this as (10). 2 times

• 5. Repeat step 3 but take 25 pennies. Record the mean age as (25). 1 time

February 4, 2016Objectives:1. Sampling Distribution of quantifiable data

(mean).2. Central Limit Theorem.

• Population has a normal distribution = the sampling distribution is normal regardless of sample size.

• Any population shape = using a small sample size the sampling distribution will have similar shape.

• Any population shape = Large sample size (n>30), the sampling distribution will be approx. normal.

The composite scores of students on the ACT college entrance examination in a recent year had a Normal distribution with mean µ = 20.4 and standard deviation = 5.8. 1. What is the probability that a randomly chosen

student scored 24 or higher on the ACT?

2. What are the mean and standard deviation of the average ACT score for an SRS of 30 students?

3. What is the probability that the average ACT score of an SRS of 30 students is 24 or higher?