Sampling Distribution of a sample Means

Section 7.3

Let’s finish the class height….

Population (5, 10, 15, 20, 25)…all samples of

size 2 – mean.

What should a sampling distribution for rolling 1 dice 20 times look like?

What should a sampling distribution for the mean when rolling 2 dice 20 times look like?

What should a sampling distribution for the mean when rolling 10 dice 20 times look like?

Simulation of Central Limit Theorem http://onlinestatbook.com/simulations/CLT/

clt.html

*Averages are less variable than individual observations!

Central Limit Theorem

When n is sufficiently large, the sampling distribution of is well approximated by a normal curve, even if the population distribution is not itself normal. (if n>30)

x

To clarify:

A sample mean ( ) based on a large n will tend to be closer to the population mean ( ) than one based on a small n.

The larger the sample size, n, the smaller the standard deviation.

X

X

Properties of a Sampling Distribution of Sample Means

Shape is approximately normal since Population is normal, or Central Limit Theorem (n 30)

x

x n

If a population has a mean equal to 52 and a standard deviation equal to 3.4 and n = 100

What is the mean of the sample mean?

What is the standard deviation of the sample mean?

Describe the shape of the sampling distribution.

Thus…...

x

x

value meanz

st dev

xz

xz

n

The amount of spaghetti sauce that fills a glass jar is normally distributed with and Find the

probability that A jar, selected at random, is between 848 g. and 855 g.

A sample of 24 jars has a mean weight between 848 and 855 grams.

850 .g 8 .g

The heights of kindergarteners are approximately normal with and . A class of 30 is used as a sample. What is the probability that the sample mean is between 38 and 40?

39in 2in

What would be different if I only asked the probability of finding a randomly selected kindergartner who is between 38 and 40 inches?

The mean of song duration is and . If the population is normally distributed, find the probability that in a sample of 30 songs, the duration is more than 14.5 seconds?

13.8sec 11.8sec

A soft drink as an average of 12 oz. Suppose x is normally distributed with standard deviation of 0.16 oz. If 16 cans are randomly selected and the volume is checked, what’s the probability that the average of the sample is more than 12.5 oz?

The composite scores of individual students on the ACT college entrance examination in 2009 followed a Normal distribution with mean 21.1 and standard deviation 5.1.

What is the probability that a randomly chosen student scored 23 or higher?

What is the probability that a SRS of 50 students had a mean score of 23 or higher?

Homework

Worksheet on SD of Means