# University of Washington Bothell Emergency Manager Core Competencies Discussion

The Module for Week 1 contains a spreadsheet showing all the Washington State DOT contract prices for 3/8-inch hot mix asphalt mixtures from 25 September 2017 to 25 September 2018. Use this data to answer this question.

Plot a histogram of this data (45 points) using the guidance in the course notes (both lectures are attached). Based on your plot, how do you characterize the data?

– uniformly distributed?

– normally distributed?

– bimodal?

Here is the raw data download from the Washington State DOT showing the low bid prices for 3/8-inch HMA over a year’s pe

This data are organized a bit better on the next tab for your analysis.

Use this data to answer quiz questions in week 1.

Bid Item Unit Price Tabulation Standard Items

Standard Item Number

Region: State Wide

Contracts awarded from 9/25/2017 thru 9/25/2018

Unit Of Measure Item Description

Northwest

5766

TON

HMA CL. 3/8 IN. PG 64-22

5766

TON

HMA CL. 3/8 IN. PG 64-22

5766

TON

HMA CL. 3/8 IN. PG 64-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

Northwest Region Average Low Bid for Item No. 5766 is $141.52 Total Quantity Planned: 37,570.00 Total Amount: $5

State Average Low Bid for Item No. 5766 is $96.06 Total Quantity Planned: 283,714.78 Total Amount: $27,253,838.05

North Central

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

North Central Region Average Low Bid for Item No. 5766 is $82.03 Total Quantity Planned: 46,313.20 Total Amount: $

State Average Low Bid for Item No. 5766 is $96.06 Total Quantity Planned: 283,714.78 Total Amount: $27,253,838.05

Southwest

5766

TON

HMA CL. 3/8 IN. PG 64-28

5766

TON

HMA CL. 3/8 IN. PG 64-28 (BRIDGE DECK)

5766

TON

HMA CL. 3/8 IN. PG 64-28 (TRANSITION)

5766

TON

HMA CL. 3/8 IN. PG 64-22

5766

TON

HMA CL. 3/8 IN. PG 58V-22

5766

TON

HMA CL. 3/8 IN. PG 58V-22 (BRIDGES)

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22 (BRIDGE)

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 58H-22 (BRIDGE)

5766

TON

HMA CL. 3/8 IN. PG 58H-22

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28 (BRIDGE)

5766

TON

HMA CL. 3/8 IN. PG 58H-22

Southwest Region Average Low Bid for Item No. 5766 is $94.25 Total Quantity Planned: 123,072.78 Total Amount: $1

State Average Low Bid for Item No. 5766 is $96.06 Total Quantity Planned: 283,714.78 Total Amount: $27,253,838.05

South Central

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

South Central Region Average Low Bid for Item No. 5766 is $89.74 Total Quantity Planned: 19,417.00 Total Amount: $

State Average Low Bid for Item No. 5766 is $96.06 Total Quantity Planned: 283,714.78 Total Amount: $27,253,838.05

Eastern

5766

TON

HMA CL. 3/8 IN. PG 76-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

5766

TON

HMA CL. 3/8 IN. PG 64H-28

Eastern Region Average Low Bid for Item No. 5766 is $83.64 Total Quantity Planned: 57,341.80 Total Amount: $4,796

State Average Low Bid for Item No. 5766 is $96.06 Total Quantity Planned: 283,714.78 Total Amount: $27,253,838.05

ces for 3/8-inch HMA over a year’s period.

e Tabulation Standard Items

m 9/25/2017 thru 9/25/2018

Total records found: 45

Job Number Contract Number AD Date

Planned Quantity

Low Bid

17A011

009161

17A015

009168

17A018

009183

17A040

009212

17A038

009220

17A043

009221

17A037

009227

17A021

009231

17A044

009256

nned: 37,570.00 Total Amount: $5,316,830.08

78 Total Amount: $27,253,838.05

09/25/2017

10/16/2017

11/20/2017

01/22/2018

02/05/2018

02/12/2018

02/20/2018

02/26/2018

03/26/2018

4.00

1,940.00

9,749.00

400.00

8,430.00

7,919.00

590.00

8,280.00

258.00

$404.52

$400.00

$145.00

$70.00

$97.50

$98.00

$140.00

$154.00

$550.00

18B002

009181

18B003

009209

18B005

009236

18B007

009260

18B009

009261

18B010

009270

18B014

009274

18B012

009275

18B011

009279

lanned: 46,313.20 Total Amount: $3,798,923.00

78 Total Amount: $27,253,838.05

11/20/2017

01/16/2018

03/05/2018

04/02/2018

04/02/2018

04/16/2018

04/23/2018

04/23/2018

04/30/2018

23,537.00

755.00

523.00

20,123.00

240.00

430.00

375.00

260.20

70.00

$85.00

$100.00

$125.00

$70.00

$200.00

$145.00

$150.00

$215.00

$375.00

11/27/2017

11/27/2017

11/27/2017

12/26/2017

02/12/2018

02/12/2018

02/26/2018

02/26/2018

02/26/2018

03/05/2018

04/09/2018

04/09/2018

04/02/2018

04/16/2018

400.00

160.00

70.00

1,670.00

65,636.00

1,226.00

29,827.50

35.49

4,563.79

670.00

4,555.00

235.00

1,830.00

10,650.00

$450.00

$423.00

$423.00

$105.00

$79.25

$100.00

$98.00

$110.00

$185.00

$160.00

$130.00

$160.00

$157.00

$80.00

17X307

17X316

17X316

16X311

18X304

18X304

18X317

18X317

18X322

18X337

18X320

18X320

18X336

18X308

009184

009185

009185

009197

009222

009222

009229

009229

009232

009238

009262

009262

009265

009268

18X308

009268

18X318

009291

ned: 123,072.78 Total Amount: $11,599,333.05

78 Total Amount: $27,253,838.05

04/16/2018

05/29/2018

144.00

1,400.00

$145.00

$110.00

18Y002

009250

18Y001

009257

18Y008

009286

lanned: 19,417.00 Total Amount: $1,742,555.00

78 Total Amount: $27,253,838.05

03/12/2018

03/26/2018

05/14/2018

7,610.00

691.00

11,116.00

$88.00

$137.00

$88.00

17Z010

009187

18Z010

009210

18Z009

009223

17Z004

009252

18Z011

009258

18Z005

009287

18Z014

009289

18Z025

009298

d: 57,341.80 Total Amount: $4,796,196.92

78 Total Amount: $27,253,838.05

12/11/2017

01/22/2018

02/20/2018

03/19/2018

03/26/2018

05/14/2018

05/21/2018

07/09/2018

20,400.00

415.00

9,730.00

11,762.80

8,040.00

6,350.00

81.00

563.00

$75.55

$135.00

$70.00

$84.33

$94.40

$86.80

$474.00

$315.00

Bid Item Unit Price Tabulation Standard Items

Region: State Wide

Contracts awarded from 9/25/2017 thru 9/25/2018

Standard Item Number

Unit Of Measure Item Description

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

5766

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

TON

HMA CL. 3/8 IN. PG 64-22

HMA CL. 3/8 IN. PG 64-22

HMA CL. 3/8 IN. PG 64-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64-28

HMA CL. 3/8 IN. PG 64-28 (BRIDGE DECK)

HMA CL. 3/8 IN. PG 64-28 (TRANSITION)

HMA CL. 3/8 IN. PG 64-22

HMA CL. 3/8 IN. PG 58V-22

HMA CL. 3/8 IN. PG 58V-22 (BRIDGES)

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22 (BRIDGE)

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 58H-22 (BRIDGE)

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28 (BRIDGE)

HMA CL. 3/8 IN. PG 58H-22

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 76-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

Total records found: 4

5766

5766

TON

TON

HMA CL. 3/8 IN. PG 64H-28

HMA CL. 3/8 IN. PG 64H-28

d Items

/2018

Total records found: 45

Planned Quantity Low Bid

4.00

1,940.00

9,749.00

400.00

8,430.00

7,919.00

590.00

8,280.00

258.00

23,537.00

755.00

523.00

20,123.00

240.00

430.00

375.00

260.20

70.00

400.00

160.00

70.00

1,670.00

65,636.00

1,226.00

29,827.50

35.49

4,563.79

670.00

4,555.00

235.00

1,830.00

10,650.00

144.00

1,400.00

7,610.00

691.00

11,116.00

20,400.00

415.00

9,730.00

11,762.80

8,040.00

6,350.00

$404.52

$400.00

$145.00

$70.00

$97.50

$98.00

$140.00

$154.00

$550.00

$85.00

$100.00

$125.00

$70.00

$200.00

$145.00

$150.00

$215.00

$375.00

$450.00

$423.00

$423.00

$105.00

$79.25

$100.00

$98.00

$110.00

$185.00

$160.00

$130.00

$160.00

$157.00

$80.00

$145.00

$110.00

$88.00

$137.00

$88.00

$75.55

$135.00

$70.00

$84.33

$94.40

$86.80

81.00

563.00

$474.00

$315.00

USE

THIS

COLUMN

Basic Statistical Measures

1 INTRODUCTION

What area of study involves taxes, war, Caesar Augustus, insurance, gambling, the Domesday

Book, Guinness, geology and pavements (to name a few of the big ones)? Well, of course, it is

statistics.

For the most part, statistical methods as we use them today are largely a product of the 20 th

century. However, that said, the origins of this topic area (the ultimate in applied mathematics)

goes back to at least the Roman Empire and further back to the Babylonians and Egyptians. As

Steel and Torrie (1960) note, the term “statistics” likely started as state arithmetic needed to

levy taxes or wage war since counts of money and subjects are needed to do both. Have we

not been reminded from time-to-time (at least one a year) that Caesar Augustus decreed that

all the world should be taxed (an interesting concept). In 1086, William the Conqueror ordered

a statistical survey of England. This was an attempt to measure the land related wealth of the

country. The recorded volumes became known as the Domesday Book (a corruption of

Doomsday since the assessment of levies / taxes were irrevocable).

By the 14th century, Flemish shipping was using an early form of probability for ship insurance

(some might say it was really more a form of gambling). This appears to have evolved into the

current insurance industry (one wonders if Lloyd’s needs a refresher course in probability).

More specifically, Pascal (1623 – 1662) the French philosopher, mathematician, and physicist

and Fermat (1601-1665) also a French mathematician together developed the theory of

probability in 1654.

The normal distribution (or curve) was first published by de Moivre in 1733 but remained

unknown until his work was discovered in a library 200 years later (one must wonder if any of

our reports will enjoy the same fate). However, more well known was the “discovery” of the

normal distribution by Laplace (1749 – 1827), a French astronomer and mathematician, and

Gauss (1777 – 1855), a German mathematician. In fact, Gauss also developed the method of

least squares which we hold so dear in regression analysis. The normal curve is often referred

to as the “Gaussian Curve” (and rightfully so).

In the period 1830 – 1833, Charles Lyell (a Scottish geologist (1797 – 1875)) published three

volumes entitled Principles of Geology. He helped to identify layers of rocks according to fossils

(and associated fossil counts). He counted the strata names based on the proportions of fossils

as Pleistocene (most recent), Pliocene (majority recent), Miocene (minority recent) and Eocene

(dawn of the recent). “Recent” referred to live mollusks he observed in the oceans. The older,

deeper strata contained fewer fossils of living species. Apparently, Charles Darwin was a fan of

Lyell and much of Darwin’s eventual work was statistically based.

1

Other well-known statisticians include Pearson (1857 – 1936, British mathematician) and W.S.

Gosset (1876 –1937). Gosset was a student of Karl Pearson and was employed by Guinness

brewers. Apparently, Gosset had a pseudonym of “student” while working at Guinness. He

developed the “t-test” hence it often referred to as “Student’s t”.

There are many others such as R.A. Fisher who introduced the concept of analysis of variation

which is basically “an arithmetic process for partitioning a total sum of squares into

components associated with recognized sources of variations” (after Steel and Torrie 1960).

2 DEFINITIONS

POPULATION (Willenbrock 1976). All measurements or counts that are obtainable from all of

the objects that process some common characteristic. Example: a “population” of data would

be the pavement condition measured on all Interstate highways in a specific state.

SAMPLE (Willenbrock 1976). A set of measurements or counts that constitute a part (or all) of

the population.

RANDON SAMPLING (Willenbrock 1976). A sampling procedure whereby any one measurement

in the population is as likely to be included as any other.

BIASED SAMPLING (after Willenbrock 1976). A sampling procedure whereby certain individual

measurements have a greater chance of being included than others. Example: biased sampling

would be taking density measurements only at places on a base course that appeared to be

well compacted.

MEAN. Average of a group of measurements. The population mean is designated (Greek mu)

x mean by σ (Greek sigma).

and a sample

MEDIAN. The number, in a set of numbers arranged in ascending order, that divides the set so

that half of the numbers are higher and half are lower.

RANGE. The largest measurement minus the smallest measurement in a group of data.

STANDARD DEVIATION. A measure of variation or dispersion of a group of data. Specifically,

the average of the squares of the numerical differences of each measurement (or observation)

from the mean. The population standard deviation is designated by “σ” and a sample standard

deviation by “s.”

HISTOGRAM. A graphical form of data presentation. A bar chart that shows in terms of area the

relative number of measurements of different classes. The width of the bar represents the

class interval, the height represents the number of measurements.

VARIABLE (ASTM 1985). A quantity to which any of the values in a given set may be assigned,

i.e., something on which measurements are made.

2

3 HISTOGRAM CONSTRUCTION

3.1 NUMBER OF CELLS (K) AFTER BLANK (1980))

The text has its own recommendations but they are rather complex mathematically. These are

simpler:

Use 10 to 20 cells for >50 data points

Use 6 to 10 cells for 1474 kg/ m 3 )

= 1.0000 – 0.0047

= 0.9953 or 99.53 percent

This makes sense: 1471 kg/m3 is way to the left of the proctor density plot (Figure 6) so you

would expect there to be a high probability that the density will be greater than 1471 kg/m 3.

2.2 2.2.2 EXAMPLE 2: PORTLAND CEMENT CONCRETE STRENGTH

If a distribution of PCC strength data is μ = 40 MPa and σ = 5.0 MPa, answer the following

questions:

a.

b.

c.

d.

What is the probability strength will be less than or equal to 50.0 MPa?

What is the probability strength will be less than or equal to 40.0 MPa?

What is the probability strength will be more than 30.0 MPa?

What is the probability strength will be less than or equal to 30.0 MPa?

Refer to Figure 8 for the results.

10

Figure 8. PCC strength probabilities.

From Table 1

z 30

30 40

2 (area less than Z = -2 equals 0.0228)

5.0

z 40

40 40

0 (area less than Z = -0 equals 0.5000)

5.0

z 50

50 40

2 (area less than Z = +2 equals 0.9772)

5.0

(a) P (strength < 50 Mpa) = 1.000 – 0.0228 = 0.9772 or 97.72%
(b) P (strength < 40 Mpa) = 1.000 – 0.5000 = 0.5000 or 50.00%
(c) P (strength > 30 Mpa) = 1.000 – 0.0228 = 0.9772 or 97.72%

(d) P (strength < 30 Mpa) = 0.0228 or 2.28%
11
2.3 EXAMPLE 3: READY-MIX CONCRETE
The contractor claims that the batch plant can produce PCC mix with
μ 35.0 MPa
σ 3.0 MPa
Assume that a somewhat unrealistic job specification states that an acceptable PCC must have a
compressive strength no lower than 30.0 MPa and no higher than 40.0 MPa (after seven days of
cure).
Question: If the contractor sends 50 truckloads of this mix to the job site, how many of the
trucks should be rejected if you know the real, potential compressive strength of each truckload
(of course you cannot do this but what the heck)?
Solution: Using z-statistic tables, you find that the total area under the standard normal
distribution between 30.0 and 40.0 MPa is about 0.90 (recall that the maximum is 1.0000 under
the curve).
Thus, approximately 90 percent of the population will be between 30.0 and 40.0 Mpa. Thus,
1.0000 – 0.90 = 0.10 or about 10 percent of the 50 trucks (i.e. about 5 trucks) should be
rejected.
Try to match this solution by using Table 1. Hint: start by computing z 30 and z 40 , then use
Table 1
2.4 EXAMPLE 4: AC BINDER CONTENTS
Assume that the AC binder data in the introduction reading represents a population, i.e.,
x μ
s σ
What is the probability that the binder content will be greater than 5.0% and 5.3% for Mix A?
Less than 6.0% and 6.3% for Mix B?
Solution
Mix A: x 5.04% μ
s 0.18% σ
z
x μ 5.0 5.04
0.22
σ
0.18
12
z
5.3 5.04
1.44
0.18
P(binder > 5.0% = 1.000 – 0.4129 = 0.5871 or 59%

P(binder > 5.3% = 1.000 – 0.9251 = 0.0749 or 7%

Mix B: x 6.09% μ

s 0.18% σ

z

6.0 6.09

0.50

0.18

z

6.3 6.09

1.167

0.18

P(binder < 6.0%) = 0.3085 or 31%
P(binder < 6.3%) = 0.8784 or 88%
or
P(binder > 6.3% = 1.0000 – 0.8784 = 0.1216 or 12%

13

(use Table 1)

3 CENTRAL LIMIT THEOREM

The Central Limit Theorem (CLT) allows use of a normal distribution to make inferences about

the population from a non-normal sampling distribution.

Central Limit Theorem

“The population may have any unknown distribution with a mean and a variance 2 . Take

samples of size n from the population. As the size n increases, the distribution of sample means

will approach a normal distribution with mean and a variance 2 /n”

x for any μ ,σ 2

x N μ,σ

as n

2

x

increases

where σ x = σ

n

=> standard deviation of the mean

Now, we can work with a distribution of sample averages x , not x.

z

x μ

σ

n

3.1 EXAMPLE – AC BINDER CONTENTS

Mix A:

n 10

x 5.3%

assumeμ 5.0%

s 0.18%

assume σ

What is the chance that the average binder content 5.3% ?

z

5.3 5.0

0.18 10

5.270

Probbinder mean 5.3% 1.000 1.000 0%

What if s is doubled, i.e., σ 0.36%?

5.3 5.0

2.635 Pbinder mean 5.3% 1.000 0.9958 0.0042 0.4%

0.36

10

What if s is multiplied by 4, i.e., σ 0.72% ?

z

14

5.3 5.0

1.318 Pbinder mean 5.3% 1.000 0.9063 0.0937 9%

0.72

10

What if n 5, not10 :

z

z

z

z

5.3 5.0

3.73 Pbinder mean 5.3 1.000 0.999 0.0001 0.01%

0.18

5

5.3 5.0

1.86 Pbinder mean 5.3 1.000 .9686 0.0314 3.1%

0.36

5

5.3 5.0

0.93 Pbinder mean 5.3 1.000 0.8238 0.1762 18%

0.72

5

4 THE T-DISTRIBUTION

4.1 WHY USE THE T-DISTRIBUTION?

When using inferential statistics we often have to decide which standard statistic to use.

Previously we have discussed the standard normal distribution (remember that in such a

distribution the average is z = 0, and the standard deviation is 1). That is convenient because

with a simple mathematical manipulation we can turn any normal distribution into the standard

normal distribution. It is advantageous because all the probabilities (area under the curve) for

the standard normal distribution have already been worked out and therefore we don’t have to

concern ourselves with taking integrals of the normal distribution equation (or remembering to

buy batteries for our fancy calculator or buy some lame calculator app in the app store).

Remember this equation to obtain the z-statistic for a sample mean:

𝑧𝑐𝑎𝑙𝑐 =

(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛) − (ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑚𝑒𝑎𝑛)

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

To use this equation to make inferences about the population based on a sample we need to

know 3 things:

1. Sample mean. Pretty straightforward, we can get that if we have the sample values.

2. Hypothesized value of the population mean. This usually comes something we are testing.

𝜎

3. Standard error. This is calculated as 𝑛. This seems to be okay: n is sample size easy to

√

obtain) BUT σ is the population standard deviation. Often, we don’t know this.

One might ask, “how can we estimate population parameters if we have to know one of them,

the standard deviation, in order to make estimates?” There are 2 answers to that.

15

1. It may be that the population is well-known and that there exists (usually in the literature) a

fairly good estimate of the population standard deviation. Wishful, but possible.

2. It may not be known at all. If this is the case, we have to estimate the population standard

deviation using the best information we have: the sample standard deviation.

If #2 is what we choose then the standard error equation becomes:

𝑠

√𝑛

. This looks identical to

the previous standard error equation except that we substituted the SAMPLE standard

deviation (s) for the POPULATION standard deviation (σ). But the sample standard deviation is

only an estimate of the population standard deviation and is not likely to be a perfect estimate.

There must be some penalty in accuracy to pay for using such a gross estimate.

There is a penalty to pay…so to speak. We no longer get to use the standard normal

distribution to figure out probabilities. Instead, we use a slightly different distribution called

“Student’s t distribution” or simply the “t distribution”. The t distribution looks very much like

the standard normal distribution for large sample sizes but as sample size gets smaller the t

distribution has noticeably fatter tails than the standard normal distribution. In other words,

there is more area under the curve at the tails in the t distribution. What happens then is that if

we use the t distribution instead of the standard normal distribution our resulting confidence

intervals (covered in the next lesson) become larger meaning that we are less certain of our

conclusion. That’s our penalty.

In other words: The t-distribution is used to test sample means when the population variance

and population standard deviation are not known. The “t-statistic” is quite similar to the “zstatistic” but also includes consideration of the sample size (n).

Where

z

t

x μ

σ

(this is the standard normal z-statistic)

x μ

s

n

(this is the equivalent t-statistic)

Table 2 is used to provide a partial listing of t-values.

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Table 2. Partial t-Distribution Table for Two-Tailed Test (after Steel and Torrie 1960)

Degrees

of

Freedom

Probability of a larger t

1

2

3

4

5

6

7

8

9

10

15

20

25

30

0.10

0.05

0.02

0.01

6.314

2.920

2.353

2.132

2.015

1.943

1.895

1.860

1.833

1.812

1.753

1.725

1.708

1.697

1.645

12.706

4.303

3.182

2.776

2.571

2.447

2.365

2.306

2.262

2.228

2.131

2.086

2.060

2.042

1.960

31.821

6.965

4.541

3.747

3.365

3.143

2.998

2.896

2.821

2.764

2.602

2.528

2.485

2.457

2.326

63.657

9.925

5.841

4.604

4.032

3.707

3.499

3.355

3.250

3.169

2.947

2.845

2.787

2.750

2.576

Probability is one-half for a one-tail test, e.g. if df = 4 and = 5% for one-tail, then t = 2.132

(i.e., use column with P = 10% for two-tail condition).

This concept will be illustrated in more detail in the next paragraph.

(a)

How is the t-distribution formed?

determine sample size n

take one sample (size n) and compute x and s 2

Calculate t

x μ

s

n

Where s

= standard deviation of sample means

n

think of t as a conversion factor to obtain probabilities

plot t on a histogram of t values

Repeat for k samples

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Example-AC Binder Contents

Mix A n = 10

x = 5.04%

S = 0.18%

18

What is the chance that the average binder content 5.2%?

t calc

x -μ

s

n

(assume μ = 5.0%)

5.2 5.0

3.51

0.18

10

One-tail test and degrees of freedom (v) = n-1 = 9

t table From Table 2, v = 9, shows α = 0.01

for t = 3.250, thus Prob