# STAT 1450 Lecture 15: Chapter 15 Notes - Sampling Distributions

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Published on 17 Sep 2020

Department

Statistics

Course

STAT 1450

Professor

Chapter 15, page 1

STAT 1450 COURSE NOTES – CHAPTER 15

SAMPLING DISTRIBUTIONS

Connecting Chapter 15 to our Current Knowledge of Statistics

Chapters 1 9 focused a lot on data analysis and data collection.

Chapter 12 introduced us to probability.

A primary tenet was that if we obtained a representative sample of some phenomenon,

then the rules of probability would allow us to generalize information from our random

sample to make a statement about our population of interest.

Using Your Intuition

Suppose you generate a random sample of 9 values (0 to 10); yielding an average of 6.7.

The instructor then collects your average along with the averages of 4 other students.

The instructor cme he aeage f he aeage.

Which data set would you expect to have a larger range? ____________

a) Your sample of 9 values or b) The instructor e f 5 averages

Then, which would you expect to have a greater likelihood of occurring? ___________

a) You obtaining an average greater than 8.

b) The instructor obtaining an average of averages greater than 8

In this chapter we will support your intuition

with rules that describe how sample statistics behave.

15.1 Parameters and Statistics

We will use ____________________________ to estimate __________________________.

From Chapter 8.

A _____________ is a number that describes a characteristic of the _______________

(e.g., the average number of text messages sent yesterday by all Ohio State students).

Often the value of a parameter is unknown because we cannot examine the entire population.

A ____________ is a number that can be computed from the __________ data.

(e.g., the average number of text messages sent yesterday by a random sample of OSU juniors).

samplestatistics populationparameters

parameter population

statistic sample

Chapter 15, page 2

Notation:

___

population mean

___

population standard deviation

x

sample ________

___

sample standard deviation

15.2 Statistical Estimation and the Law of Large Numbers

If we want to estimate the population mean μ, we use the sample mean

x

. The sample mean

x

is a random variable. We learned in Chapter 12 that each random variable has a probability

model which tells us the values the random variable can take and the probability that it takes on

these values.

Hee an ineeing hgh: he babili ha he amle mean

x

is exactly equal to the

population mean μ is 0. However, we expect the sample mean to be somewhere near the

population mean.

x

is an unbiased estimator of μe dn eec i e- or underestimate

the population mean.

And, as we sample more and more individuals from the population, we expect the sample mean

to get closer to the population meanthis is a Law of Large Nmbe

Law of Large Numbers

Draw observations at random from any population with finite mean μ.

As the number of observations drawn increases,

the mean

x

of the observed values gets closer and closer to the mean μ of the population.

Check out Example 15.3 on page 347 of the book for a nice example and graphic of the law of

large numbers.

15.3 Sampling Distributions

The law of large numbers says,

sample enough individuals and the _________

x

__________ the unknown ____________.

Typically we take just one sample and then generalize to the population as a whole. Before we

do that, we need to understand how

x

behaves. This can be done through simulation

(see Example 15.4 on page 350).

Poll: As the sample size increases, the variability of the mean,

x

, of the observed values ______.

a) increases.

b) decreases.

no

s

statistic nitproach parameter

O

## Document Summary

Connecting chapter 15 to our current knowledge of statistics. Chapters 1 (cid:177) 9 focused a lot on data analysis and data collection. A primary tenet was that if we obtained a representative sample of some phenomenon, then the rules of probability would allow us to generalize information from our random sample to make a statement about our population of interest. Suppose you generate a random sample of 9 values (0 to 10); yielding an average of 6. 7. The instructor then collects your average along with the averages of 4 other students. The instructor c(cid:82)m(cid:83)(cid:88)(cid:87)e(cid:86) (cid:87)he (cid:181)a(cid:89)e(cid:85)age (cid:82)f (cid:87)he a(cid:89)e(cid:85)age(cid:86). (cid:182) Which data set would you expect to have a larger range: your sample of 9 values or, the instructor(cid:182)(cid:86) (cid:86)e(cid:87) (cid:82)f 5 averages. Then, which would you expect to have a greater likelihood of occurring: you obtaining an average greater than 8, the instructor obtaining an average of averages greater than 8 (cid:171)