powerpoint sampling distribution

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.

SAMPLING DISTRIBUTIONS

8-1


SAMPLING DISTRIBUTION OF THE SAMPLE MEANS: OBJECTIVES

1. Describe the distribution of the sample mean: normal population

2. Describe the distribution of the sample mean: nonnormal population

8-2


DISTRIBUTION OF COINS

Histogram of coins collected over 6 months.

8-3


SIMULATION

Use the coin distribution to find samples and take the mean of each sample. Now take all the possible samples (through simulation) and put the sample means ( ) in a distribution.

This is called The Sampling Distribution of the Mean

8-4

x


SIMULATION

Top : Distribution of Coins

Middle:One Sample

Bottom:Distribution of sample means

8-5


Statistics such as are random variables since their value varies from sample to sample.

As such, they have probability distributions associated with them.

8-6

x


The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n.

The sampling distribution of the sample mean is the probability distribution of all possible values of the random variable computed from a sample of size n from a population with mean μ and standard deviation σ.

8-7

x x


8-8

Illustrating Sampling Distributions

Step 1: Obtain a simple random sample of size n.


8-9



Step 2: Compute the sample mean.


8-10



Step 2: Compute the sample mean.

Step 3: Assuming we are sampling from a finite population, repeat Steps 1 and 2 until all simple random samples of size n have been obtained.


OBJECTIVE 1 Describe the Distribution of the Sample Mean:

Normal Population

8-11


What role does n, the sample size, play in the standard deviation of the distribution of the sample mean?

8-12


What role does n, the sample size, play in the standard deviation of the distribution of the sample mean?

8-13

As the size of the sample increases, the standard deviation of the distribution of the sample mean decreases.


8-14

Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of will have mean and standard deviation

The standard deviation of the sampling distribution

of is called the standard error of the mean andis denoted .

The Mean and Standard Deviation of theSampling Distribution of

x

x

x

x n

.

x

x


If a random variable X is normally distributed, the distribution of the sample mean is normally distributed.

8-15

The Shape of the Sampling Distribution of If X is Normal x

x


OBJECTIVE 2 Describe the Distribution of the Sample Mean: Nonnormal

Population

8-16


The mean of the sampling distribution is equal to the mean of the parent population and the standard deviation of the sampling distribution of the sample mean is regardless of the sample size.

The Central Limit Theorem: the shape of the distribution of the sample mean becomes approximately normal as the sample size n increases, regardless of the shape of the population. 8-17

Key Points

n


8-18

Parallel Example 5: Using the Central Limit Theorem

Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes.

(a) If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean.

(b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes?


8-19




Solution: is approximately normally distributed with mean = 11.4 and std. dev. = .

3.235

0.5409x


8-20




(b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes?

Solution: is approximately normally distributed with mean = 11.4 and std. dev. = .

Solution: Using StatCrunch with above, P = 0.23.

3.235

0.5409x


SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTION: OBJECTIVES

1. Describe the sampling distribution of a sample proportion2. Compute probabilities of a sample proportion

8-21


OBJECTIVE 1 Describe the Sampling Distribution of a Sample Proportion

8-22


POINT ESTIMATE OF A POPULATION PROPORTION

Suppose that a random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The sample proportion, denoted (read “p-hat”) is given by

where x is the number of individuals in the sample with the specified characteristic. The sample proportion is a statistic that estimates the population proportion, p.

8-23

ˆ p

ˆ p xn

ˆ p


In a Quinnipiac University Poll conducted in May of 2008, 1745 registered voters nationwide were asked whether they approved of the way George W. Bush is handling the economy. 349 responded “yes”. Obtain a point estimate for the proportion of registered voters who approve of the way George W. Bush is handling the economy.

8-24

Parallel Example 1: Computing a Sample Proportion


In a Quinnipiac University Poll conducted in May of 2008, 1,745 registered voters nationwide were asked whether they approved of the way George W. Bush is handling the economy. 349 responded “yes”. Obtain a point estimate for the proportion of registered voters who approve of the way George W. Bush is handling the economy.

8-25

Parallel Example 1: Computing a Sample Proportion

Solution: p̂ 349

17450.2


According to a Time poll conducted in June of 2008, 42% of registered voters believed that gay and lesbian couples should be allowed to marry.Describe the sampling distribution of the sample proportion for samples of size n = 10, 50, 100.

Note: We are using simulations to create the histograms on the following slides.

8-26

Parallel Example 2: Using Simulation to Describe the Distribution of the Sample Proportion


Shape: As the size of the sample, n, increases, the shape of the sampling distribution of the sample proportion becomes approximately normal.

Center: The mean of the sampling distribution of the sample proportion equals the population proportion, p.

Spread: The standard deviation of the sampling distribution of the sample proportion decreases as the sample size, n, increases.

8-27

Key Points


For a simple random sample of size n with population proportion p:

• The shape of the sampling distribution of is approximately normal provided np(1 – p) ≥ 10.

• The mean of the sampling distribution of is

• The standard deviation of the sampling distribution of is

8-28

Sampling Distribution of

ˆ p

ˆ p

ˆ p p

ˆ p

ˆ p p(1 p)

n

p̂


• The model on the previous slide requires that the sampled values are independent. When sampling from finite populations, this assumption is verified by checking that the sample size n is no more than 5% of the population size N (n ≤ 0.05N).

• Regardless of whether np(1 – p) ≥ 10 or not, the mean of the sampling distribution of is p, and the standard deviation is

8-29

Sampling Distribution of

ˆ p

ˆ p p(1 p)

n

p̂


According to a Time poll conducted in June of 2008, 42% of registered voters believed that gay and lesbian couples should be allowed to marry. Suppose that we obtain a simple random sample of 50 voters and determine which voters believe that gay and lesbian couples should be allowed to marry. Describe the sampling distribution of the sample proportion for registered voters who believe that gay and lesbian couples should be allowed to marry. 8-30

Parallel Example 3: Describing the Sampling Distribution of the Sample Proportion


The sample of n = 50 is smaller than 5% of the population size (all registered voters in the U.S.). Also, np(1 – p) = 50(0.42)(0.58) = 12.18 ≥ 10.The sampling distribution of the sample proportion is therefore approximately normal with mean=0.42 and standard deviation =

(Note: this is very close to the standard deviation of 0.072 found using simulation in Example 2.) 8-31

Solution

0.42(1 0.42)50

0.0698


OBJECTIVE 2 Compute Probabilities of a Sample Proportion

8-32


According to the Centers for Disease Control and Prevention, 18.8% of school-aged children, aged 6-11 years, were overweight in 2004.(a)In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight?(b)Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

8-33

Parallel Example 4: Compute Probabilities of a Sample Proportion


• n = 90 is less than 5% of the population size• np(1 – p) = 90(.188)(1 – .188) ≈ 13.7 ≥ 10• is approximately normal with mean=0.188

and standard deviation =

(a) In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight?

(b) Use StatCrunch with mean = .188, standard deviation as calculated to be 0.0412, then the probability is 0.0485

8-34

Solution

ˆ p

(0.188)(1 0.188)90

0.0412


• is approximately normal with mean = 0.188 and standard deviation = 0.0412

(b) Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

8-35

Solution

ˆ p

Using StatCrunch the probability = 0.028.We would only expect to see about 3 samples in 100 resulting in a sample proportion of 0.2667 or more. This is an unusual sample if the true population proportion is 0.188.

p̂ 2490

0.2667