# STA 205 Lecture Notes - Lecture 8: Sampling Distribution, Standard Deviation, Flying Pig Marathon

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Packet 8: Sampling Distribution of the Sample Mean Textbook pages: 24 – 30; 701 – 704

After completing this material, you should be able to:

• explain what the symbols

y

µ

and

y

σ

represent.

• describe the sampling distribution of the sample mean by discussing its shape, mean, and standard deviation.

• find probabilities associated with various sample means based on the sampling distribution.

• make inferences from the probability and explain the reasoning.

A local pub is interested in learning more about their daily sales. In order to do this, daily sales (measured in dollars) from

the past two months are collected. A histogram and summary statistics are given below. Use these to fill in the description

of the distribution below – include units with the values where appropriate.

The distribution of ______________________________________ has a shape

that is ________________________________. The average sales for these

_________ days is ________________________, and 50% of the time daily

sales are ______________________or less. The smallest sales amount was

___________________, while the largest was _____________________.

The middle 50% of daily sales fell between ______________________ and _______________________. According to

_____________________________ Rule, we expect daily sales to fall between ________________________ and

__________________________ (three standard deviations from the mean) ___________% of the time.

Let’s assume this is our population of sales values – what happens if we start to take samples of size 14 from this population

of values?

Below is the sample which was taken – the values selected in the sample

have been highlighted in pink on the original histogram.

Sample Sales:

2062

2142

2503

2566

2814

2827

3189

3446

3748

4908

5788

5816

6105

6799

What is the mean for this sample of 14 days?

If we took another sample of 14 days would we expect the same

mean?

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weekday

daily pub sales wknd

µHis same

1bimodal 2right foranytype

skewd

GO 3917.443 251 25 251,251

3611.45 TI

1210.50121 1068.2013

2853.55141 5318.05151

Chebyshev's 0

8454.3646 golor more

391744331311512.307

Hneg _O 619 478 8454.3646

if 14924542

5174113693.4854

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false to the

Fading

For All

If we take another sample

we get different sale amounts

thus we get adifferent

value of J

Asampling distribution describes

how astatistic changes in

value from sample to sample

The sample mean Tis a

statistic that summarizes

QUANTITATIVE data The

sampling distribution of g

describes what values of g

we expect igIFr sles

page 2

STA 205 Notes Buckley Fall 2018

Recall: What is a sampling distribution and why is it important?

We need to understand how the samples vary. To do this, we need to describe the sampling distribution of the sample

mean. To do this, the following three characteristics must be addressed:

•

•

•

Example: Each of the following histograms represents a sampling distribution of the sample mean for various sample

sizes. Samples of size 5, 15, and 40 were taken from some population. Match each histogram to the appropriate sample

size.

Based on the sampling distributions shown, what is the (approximate) population mean? Explain.

The population from which these samples were taken must have had what shape? Explain.

sample size: ___________

sample size: ___________

sample size: ___________

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we expect to get from samples

Shape nsolr

mean it

standard donation to

ib pieris

n5n15 n40

formally distributed