# A review of key statistical concepts

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```					A review of key statistical
concepts
An overview of the review
•   Populations and parameters
•   Samples and statistics
•   Confidence intervals
•   Hypothesis testing
Populations and Parameters

… and Samples and Statistics
Populations and Parameters
• A population is any large collection of
objects or individuals, such as Americans,
students, or trees about which information is
desired.
• A parameter is any summary number, like
an average or percentage, that describes the
entire population.
Parameters
• Examples:
– population mean µ = average temperature
– population proportion p = proportion approving
of president’s job performance
• 99.999999999999….% of the time, we
don’t (...or can’t) know the real value of a
population parameter.
• Best we can do is estimate the parameter!
Samples and Statistics
• A sample is a representative group drawn
from the population.
• A statistic is any summary number, like an
average or percentage, that describes the
sample.
Statistics
• Examples
– sample mean (“x-bar”)
– sample proportion (“p-hat”)
• Because samples are manageable in size, we
can determine the value of statistics.
• We use the known statistic to learn about
the unknown parameter.
Example: Smoking at PSU?

Population of      What proportion
42,000 PSU students
smoke regularly?

Sample of          43% reported
987 PSU students      smoking regularly
Population of
5 million college      Is the average
students           GPA 2.7?

How likely is it that
100 students would
have an average
Sample of          GPA as large as 2.9
100 college students
if the population
average was 2.7?
Example: A linear relationship?
Regression Plot
Weight = -2037.00 + 130.817 Gestation

S = 167.327   R-Sq = 77.5 %     R-Sq(adj) = 76.8 %

3500
Birth weight (grams)

E(Y) = A + B X

3000
Y-hat = a + b X

2500

34   35    36      37       38       39       40        41     42

Gestation (weeks)
Two ways to learn
• Confidence intervals estimate parameters.
– We can be 95% confident that the proportion of
Penn State students who have a tattoo is between
5.1% and 15.3%.
• Hypothesis tests test the value of parameters.
– There is enough statistical evidence to conclude
that the mean normal body temperature of adults
is lower than 98.6 degrees F.
Confidence intervals

A review of concepts
The situation
• Want to estimate the actual population
mean .
• But can only get “x-bar,” the sample mean.
• Use “x-bar” to find a range of values,
L<<U, that we can be really confident
contains .
• The range of values is called a “confidence
interval.”
Confidence intervals
for proportions in newspapers
• “Sample estimate”: 69% of 1,027 U.S. adults
think using a hand-held cell phone while driving a
car should be illegal.
• The “margin of error” is 3%.
• The “confidence interval” is 69% ± 3%.
• We can be really confident that between 66% and
72% of all U.S. adults think using a hand-held cell
phone while driving a car should be illegal.

Source: ABC News Poll, May 16-20, 2001
General form of
most confidence intervals
•   Sample estimate ± margin of error
•   Lower limit L = estimate - margin of error
•   Upper limit U = estimate + margin of error
•   Then, we’re confident that the value of the
population parameter is somewhere
between L and U.
(1-α)100% t-interval
for population mean 
Formula in words:
Sample mean ± (t-multiplier × standard error)

Formula in notation:
   
xt   
 2

 s

 1  , n  1







n



Determining the t-multiplier

0.4

0.3
density

0.2

1
0.1
                              
2                              2
0.0

-4   -3   -2   -1    0      1   2   3   4

t(14)
Typical t-multipliers
Conf. coefficient    Conf. level            
1
1           1     100 %        2

0.90              90%            0.95

0.95              95%            0.975

0.99              99%            0.995
t-interval for mean in Minitab

One-Sample T: FVC

Variable   N    Mean    StDev SE Mean    95.0% CI
FVC        8   3.5875   0.1458 0.0515 (3.4655,3.7095)

We can be 95% confident that the mean forced vital
capacity of all female college students is between 3.5
and 3.7 liters.
Length of confidence interval
• Want confidence interval to be as narrow as
possible.
• Length = Upper Limit - Lower Limit
How length of CI is affected?
   
xt s





n





•   As sample mean increases…
•   As the standard deviation decreases…
•   As we decrease the confidence level…
•   As we increase sample size …
Hypothesis testing

A review of concepts
General idea of
hypothesis testing
• Make an initial assumption.
• Collect evidence (data).
• Based on the available evidence (data),
decide whether to reject or not reject the
initial assumption.
Example: Normal body temperature

Population of
98.6 degrees? Or
is it lower?

Average body
Sample of         temperature of 130
98.25 degrees.
Making the decision
• It is either likely or unlikely that we would
collect the evidence we did given the initial
assumption.
• If it is likely, then we “do not reject” our
initial assumption. There is not enough
evidence to do otherwise.
Making the decision (cont’d)
• If it is unlikely, then:
– either our initial assumption is correct and we
experienced a very unusual event
– or our initial assumption is incorrect
• In statistics, if it is unlikely, we “reject” our
initial assumption.
Again, idea of hypothesis testing:
criminal trial analogy
• First, state 2 hypotheses, the null hypothesis
(“H0”) and the alternative hypothesis (“HA”)

– H0: Defendant is not guilty (innocent).
– HA: Defendant is guilty.
Criminal trial analogy
(continued)
• Then, collect evidence, such as finger
prints, blood spots, hair samples, carpet
fibers, shoe prints, ransom notes,
handwriting samples, etc.

• In statistics, the data are the evidence.
Criminal trial analogy
(continued)
• Then, make initial assumption.
– Our criminal justice system is based on
“defendant is innocent until proven guilty.”
– So, assume defendant is innocent.

• In statistics, we always assume the null
hypothesis is true.
Criminal trial analogy
(continued)
• Then, make a decision based on the
available evidence.
– If there is sufficient evidence (“beyond a
reasonable doubt”), reject the null hypothesis.
(Behave as if defendant is guilty.)
– If there is insufficient evidence, do not reject
the null hypothesis. (Behave as if defendant is
innocent.)
Very important point
• If we reject the null hypothesis, we do not prove
the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not
prove the null hypothesis is true.
• We merely state there is enough evidence to
behave one way or the other.
• Always true in statistics! Whatever the decision,
there is always a chance we made an error.
Errors in criminal trials

Truth
Jury
Not guilty   Guilty
Decision
Not guilty      OK        ERROR

Guilty        ERROR          OK
Errors in hypothesis testing

Truth
Null          Alternative
Decision
hypothesis      hypothesis
Do not                         TYPE II
OK
reject null                    ERROR
TYPE I
Reject null                        OK
ERROR
Definitions: Types of errors
• Type I error: The null hypothesis is
rejected when it is true.
• Type II error: The null hypothesis is not
rejected when it is false.
• There is always a chance of making one of
these errors. But, a good scientific study
will minimize the chance of doing so!
Making the decision
• “It is either likely or unlikely that we would
collect the evidence we did given the initial
assumption.”
• Two ways to determine likely or unlikely:
– Critical value approach (many textbooks)
– P-value approach (science, journals, software)

Type           Null      Alternative

Right-tailed   H0 :   3   H0 :   3

Left-tailed    H0 :   3   H0 :   3

Two-tailed     H0 :   3   H0 :   3
Critical value approach
• Using sample data and assuming null hypothesis is
true, calculate the value of the test statistic.
• Set the significance level, α, the probability of
making a Type I error to be small (0.05 or 0.01).
• Compare the value of the test statistic to the
known distribution of the test statistic.
• If the test statistic is more extreme than expected,
allowing for an α chance of error, reject the null
hypothesis. Otherwise, don’t reject the null.
Right-tailed critical value

0.4

0.3
density

0.2

0.95
0.1                                            0.05

0.0

-4   -3   -2   -1     0     1      2     3      4
t(14)       1.7613

Reject null hypothesis if test statistic is greater than 1.7613.
Left-tailed critical value
0.4

0.3
density

0.2                          0.95

0.1
0.05

0.0

-4     -3    -2   -1     0     1   2   3   4
t(14)
-1.7613

Reject null hypothesis if test statistic is less than -1.7613.
Two-tailed critical value

0.4

0.3

0.95
density

0.2

0.1
0.025                                   0.025

0.0

-4      -3     -2      -1     0     1    2       3   4

-2.1448        t(14)       2.1448

Reject null hypothesis if test statistic is less than -2.1448 or
greater than 2.1448.
P-value approach
• Using sample data and assuming null hypothesis is
true, calculate the value of the test statistic.
• Using known distribution of the test statistic,
calculate the P-value = “If the null hypothesis is
true, what is the probability that we’d observe a
more extreme test statistic than we did?”
• Set the significance level, α, the probability of
making a Type I error to be small (0.05 or 0.01).
• If the probability is small, i.e., smaller than α,
reject the null hypothesis. Otherwise, don’t reject
the null.
Right-tailed P-value

0.4

0.3
density

0.2
0.9873

0.1
0.0127

0.0

-4   -3   -2   -1     0      1   2        3     4
t(14)            t* = 2.5

If it’s unlikely to observe such a large test statistic, i.e., if the P-
value (0.0127) is smaller than α, reject the null hypothesis.
Left-tailed P-value

0.4

0.3
density

0.2                           0.9873

0.1
0.0127

0.0

-4     -3     -2   -1      0     1   2   3   4

t* = -2.5         t(14)

If it’s unlikely to observe such a small test statistic, i.e., if the P-
value (0.0127) is smaller than α, reject the null hypothesis.
Two-tailed P-value

0.4

0.3
density

0.2                         0.9746

0.1
0.0127                                0.0127

0.0

-4   -3    -2    -1     0      1   2     3      4
t* = -2.5        t(14)        t* = 2.5

If it’s unlikely to observe such an extreme test statistic, i.e., if the
P-value (0.0254) is smaller than α, reject the null hypothesis.
Example: Right-tailed test
Brinell hardness measurement of
ductile iron subcritically annealed:
170 167 174 179 179                         H 0 :   170
156 163 156 187 156
183 179 174 179 170                         H A :   170
156 187 179 183 174
187 167 159 170 179

One-Sample T: Brinell
Test of mu = 170 vs mu > 170

Variable     N    Mean      StDev      SE Mean    T       P
Brinell     25   172.52     10.31        2.06    1.22   0.117
Example: Right-tailed critical value

0.4

0.3
density

0.2
0.95

0.1                                            0.05

0.0

-4   -3   -2   -1     0     1      2       3    4
t(24)       1.7109
Example: Right-tailed P-value

0.4

0.3
density

0.883
0.2

0.117
0.1

0.0

-4   -3   -2   -1    0      1    2       3    4
t(24) t* = 1.22
Example: Left-tailed test
Height of sunflower seedlings.
11.5   11.8   15.7   16.1   14.1   10.5

H 0 :   15.7
15.2   19.0   12.8   12.4   19.2   13.5
16.5   13.5   14.4   16.7   10.9   13.0
15.1   17.1   13.3   12.4    8.5   14.3
12.9   11.1   15.0   13.3   15.8   13.5        H A :   15.7
9.3   12.2   10.3

Test of mu = 15.7 vs mu < 15.7

Variable        N     Mean         StDev   SE Mean     T       P
Sunflower      33    13.664        2.544    0.443    -4.60   0.000
Example: Left-tailed critical value

0.4

0.3
density

0.2                          0.95

0.1        0.05

0.0

-4   -3      -2   -1     0     1   2   3   4

-1.6939    t(32)
Example: Left-tailed P-value

0.4

0.3
density

0.2
>0.9999

0.1
<0.0001

0.0

-5            0      5
-4.60      t(32)
Example: Two-tailed test
Thickness of spearmint gum.
7.65   7.60   7.65   7.70     7.55
H 0 :   7.5
7.55   7.40   7.40   7.50     7.50
H A :   7.5

Test of mu = 7.5 vs mu not = 7.5

Variable N       Mean       StDev    SE Mean     T      P
Gum     10      7.5500      0.1027    0.0325   1.54   0.158
Example: Two-tailed critical value

0.4

0.3

0.95
density

0.2

0.1        0.025                            0.025

0.0

-4    -3     -2   -1     0    1   2    3      4
t(9)
-2.2622                    2.2622
Example: Two-tailed P-value

0.4

0.3
density

0.2
0.842

0.1        0.079                                   0.079

0.0

-4   -3      -2      -1     0     1      2      3    4

-1.54         t(9)   1.54

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