Section 1.

1 An overview of statistics
• Defined statistics
• Distinguished between a population and a
sample
• Distinguished between a parameter and a
statistic
• Distinguished between descriptive statistics
and inferential statistics

1
 Section 1.2 Data classification

• Distinguished between qualitative data and

quantitative data

2
 Section 1.3 Data collection and
experimental design
• Discussed sampling techniques
– simple random sampling,
– stratified sampling,
– cluster sampling,
– systematic sampling,
– convenience sampling

3
Section 2.1 Frequency distribution and
their graphs
• Constructed frequency distributions
– Limits,
– midpoints,
– relative frequencies,
– cumulative frequencies,
– and boundaries

4
 Section 2.2 More graphs and displays
• Graphed quantitative data using stem-and-leaf
plots and dot plots
• Graphed qualitative data using pie charts and
Pareto charts
• Graphed paired data sets using scatter plots
and time series charts

5
 Section 2.3 Measures of central
tendency
• Determined the mean, median, and mode of a
population and of a sample
Σx Σx
µ= x=
N n
• Determined the weighted mean of a data set and
the mean of a frequency distribution
Σ( x ⋅ w) Σ( x ⋅ f )
x= x= n = Σf
Σw n
• The mean always falls in the direction in which
the distribution is skewed.
6
 Section 2.4 Measure of variation
• Determined the range of a data set (max-min)
• Determined the variance and standard deviation
of a population and of a sample
Σ ( x − µ ) 2
Σ ( x − x ) 2
σ2 = s2 =
N n −1
• Used the Empirical Rule and Chebychev’s
Theorem to interpret standard deviation
• Approximated the sample standard deviation for
grouped data 2
Σ( x − x ) f
s=
n −1 7
 Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations

95% within 2 standard deviations

68% within 1
standard deviation

34% 34%

2.35% 2.35%
13.5% 13.5%

x − 3s x − 2s x −s x x +s x + 2s x + 3s

8
 Chebychev’s Theorem
• The portion of any data set lying within k
standard deviations (k > 1) of the mean is at least:
1
1− 2
k
• k = 2: In any data set, at least
1 3
1 − 2 =or 75%
2 4
of the data lie within 2 standard deviations of the mean.

• k = 3: In any data set, at least 1 8

1 − 2 =or 88.9%
3 9
of the data lie within 3 standard deviations of the mean.

9
 Section 2.5 Measure of position
• Determined the quartiles of a data set Q1, Q2,
Q3
• Determined the interquartile range of a data
set IQR = Q3 – Q1
• Created a box-and-whisker plot (Requires five-
number summary): Minimum entry, First
quartile Q1, Median Q2, Third quartile Q3,
Maximum entry
• Interpreted other fractiles such as percentiles
10
 Section 3.1 Basic concepts of
probability and counting
• Identified the sample space of a probability
experiment
• Used the Fundamental Counting Principle
• Determined the probability of the
complement of an event: P(E) + P(E ′) = 1

11
Section 3.2 Conditional probability and
the multiplication rule
• Determined conditional probabilities: P(B | A)
• Event A and B are independent if
– P(B | A) = P(B) or if P(A | B) = P(A)
• Used the Multiplication Rule to find the
probability of two events occurring in sequence
– P(A and B) = P(A) ∙ P(B | A) = P(B) ∙ P(A | B)
– P(A and B) = P(A) ∙ P(B) if A and B are independent

12
 Bayes’s theorem

Total probability formula:

𝑃𝑃 𝐵𝐵 = 𝑃𝑃 𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴 + 𝑃𝑃 𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴′ =𝑃𝑃 𝐴𝐴 � 𝑃𝑃 𝐵𝐵 𝐴𝐴 + 𝑃𝑃 𝐴𝐴′ 𝑃𝑃 𝐵𝐵 𝐴𝐴′ )

13
 Section 3.3 The addition rule
• Two events are mutually exclusive if A and B
cannot occur at the same time.
• Used the Addition Rule to find the probability
of two events
– P(A or B) = P(A) + P(B) – P(A and B)
– P(A or B) = P(A) + P(B) if A and B are mutually
exclusive

14
 Section 3.4 Additional topics in
probability and counting
Permutation
• An ordered arrangement of objects
• The number of different permutations of n
distinct objects is n! (n factorial)
• n! = n∙(n – 1)∙(n – 2)∙(n – 3)∙ ∙ ∙3∙2 ∙1
Permutation of n objects taken r at a time
• The number of different permutations of n
distinct objects taken r at a time
n! where r ≤ n
n Pr =
(n − r )! 15
 Distinguishable Permutations
Distinguishable Permutations
• The number of distinguishable permutations
of n objects where n1 are of one type, n2 are
of another type, and so on
■ n!
n1 !⋅ n2 !⋅ n3 !⋅ ⋅ ⋅ nk !

where n1 + n2 + n3 +∙∙∙+ nk = n

16
 Combinations
Combination of n objects taken r at a time
• A selection of r objects from a group of n
objects without regard to order
n!
n Cr =
■
( n − r )! r !

17
Section 4.1 Probability distribution
Discrete probability distribution
Lists each possible value the random variable can
assume, together with its probability.

1. The probability of each value of the 0 ≤ P (x) ≤ 1

discrete random variable is between
0 and 1, inclusive.
2. The sum of all the probabilities is 1. ΣP (x) = 1

18
 Variance and Standard Deviation
Mean of a discrete probability distribution
𝝁𝝁 = ∑ 𝒙𝒙𝒙𝒙(𝒙𝒙)

Variance of a discrete probability distribution

σ2 = Σ(x – μ)2P(x)

Standard deviation of a discrete probability distribution

σ = σ 2 = Σ( x − µ ) 2 P ( x )
Expected value of a discrete random variable
E(x) = μ = ΣxP(x)
19
Section 4.2 Binomial distribution

20
 Binomial Probability Formula
Binomial Probability Formula
• The probability of exactly x successes in n trials is
n!
= P( x) n= Cx p q x n− x
p x q n− x
(n − x)! x !
• n = number of trials
• p = probability of success
• q = 1 – p probability of failure
• x = number of successes in n trials

21
 Mean, Variance, and Standard
Deviation
• Mean: μ = np

• Variance: σ2 = npq

• Standard Deviation:
σ = npq

22
 Section 4.3 More discrete probability
distributions
Geometric distribution:
Satisfies the following conditions
A trial is repeated until a success occurs.
The repeated trials are independent of each other.
The probability of success p is constant for each trial.
x represents the number of the trial in which the first
success occurs.

23
 Geometric Distribution
• If the probability of success on each trial is p,
then the probability that the xth trial (out of x
trials) is the first success is
x −1
P( x) = pq
where
x = 1, 2,3, 4,... and q= 1− p

24
 Poisson Distribution
Poisson distribution
• A discrete probability distribution.
• Satisfies the following conditions
– The experiment consists of counting the number of
times x an event occurs in a given interval. The
interval can be an interval of time, area, or volume.
– The probability of the event occurring is the same for
each interval.
– The number of occurrences in one interval is
independent of the number of occurrences in other
intervals.

25
 Poisson Distribution
Poisson distribution
• Conditions continued:
– The probability of exactly x occurrences in an interval
is x −µ
P (x ) = e
µ
x!
where e is an irrational number ≈ 2.71828 and μ is
the mean number of occurrences per interval unit or
the probability of a rare event occurring in a given
interval.

26
Discrete
Probability X ~ B( n, p ) X ~ Geometric (p)
Distribution
Random X = the # of successful trials X = the # of the first X = the # of a rare
Variable during n success/failure successful trial even occurs
X trials
Values R.V. X = 0,1,2,…,n X = 1,2,3,… X = 0,1,2,3,…
X assumes
Parameters n = # of all trials p = P(success)

p = P(success) q=1-p=P(failure)

q=1-p =P(failure)
Probability
Formula
Mean
Formula

Variance
Formula
27
 Section 5.1 Introduction to normal
probability distribution
1. The mean, median, and mode are equal.
2. The normal curve is bell-shaped and is symmetric about
the mean.
3. The total area under the normal curve is equal to 1.

μ – 3σ μ – 2σ μ–σ μ μ+σ μ + 2σ μ + 3σ
28
 The Standard Normal Distribution
Standard normal distribution
• A normal distribution with a mean of 0 and a
standard deviation of 1.

Area = 1

z
–3 –2 –1 0 1 2 3
• Any x-value can be transformed into a z-score by
using the formula
Value − Mean x−µ
z= =
Standard deviation σ
29
 Finding Areas Under the Standard
Normal Curve
1. Sketch the standard normal curve and shade
the appropriate area under the curve.
2. Find the area by following the directions for
each case shown.
a. To find the area to the left of z, find the area that
corresponds to z in the Standard Normal Table.
2. The area to the left
of z = 1.23 is 0.8907

1. Use the table to find

the area for the z-score
30
 Section 5.2 Finding probabilities
• Found probabilities for normally distributed
variables

Section 5.3 Finding values

• Transformed a z-score to an x-value
x = μ + zσ

31
 Section 5.4 Sampling distributions and
the central limit theorem
Properties of sampling distribution of sample means:
1. The mean of the sample means, µ x , is equal to the
population mean μ.
µx = µ
2. The standard deviation of the sample means, σ x , is
equal to the population standard deviation, σ,
divided by the square root of the sample size, n.
σ
σx =
n
• Called the standard error of the mean.

32
 The Central Limit Theorem
1. Any Population Distribution 2. Normal Population Distribution

Distribution of Sample Means, Distribution of Sample Means,

n ≥ 30 (any n)

33
 Probability and the Central Limit
Theorem
• To transform x to a z-score
Value − Mean x − µ x x − µ
=z = =
Standard error σx σ
n

To find probabilities for individual member of a population

with a normally distributed random variable x, use the
formula
𝑥𝑥−𝜇𝜇
𝑧𝑧 =
𝜎𝜎
To find probabilities for the mean 𝑥𝑥̅ of a sample size n, use
the formula
̅ 𝑥𝑥�
𝑥𝑥−𝜇𝜇
𝑧𝑧 =
𝜎𝜎𝑥𝑥�
34
Section 5.5 Normal Approximation to a
Binomial
Normal Approximation to a Binomial Distribution

• If np ≥ 5 and nq ≥ 5, then the binomial random variable

x is approximately normally distributed with
– mean μ = np
– standard deviation σ = npq

35
 Using the Normal Distribution to
Approximate Binomial Probabilities
In Words In Symbols
1. Verify that the binomial Specify n, p, and q.
distribution applies.
2. Determine if you can use Is np ≥ 5?
the normal distribution to Is nq ≥ 5?
approximate x, the binomial
variable.
3. Find the mean µ and µ = np
standard deviation σ for the σ = npq
distribution.
36
 Using the Normal Distribution to
Approximate Binomial Probabilities
In Words In Symbols
4. Apply the appropriate Add or subtract 0.5
continuity correction. from endpoints.
Shade the corresponding
area under the normal
curve.
5. Find the corresponding x−µ
z=
z-score(s). σ
6. Find the probability. Use the Standard
Normal Table.
37
 Section 6.1 Constructing Confidence
Intervals for μ (σ known)
Finding a Confidence Interval for a Population Mean
(σ known)
In Words In Symbols
1. Verify that σ is known, the
sample is random, and either
the population is normally
distributed or n ≥ 30.
2. Find the sample statistics n and Σx
x=
n
x.
38
 Constructing Confidence Intervals for μ
In Words In Symbols
3. Find the critical value zc that Use the Standard
corresponds to the given Normal Table.
level of confidence.
E = zc
σ
4. Find the margin of error E.
n
5. Find the left and right Left endpoint: x − E
endpoints and form the Right endpoint: x + E
confidence interval. Interval:
x−E<µ < x+E

39
 Sample Size
• Given a c-confidence level and a margin of error
E, the minimum sample size n needed to estimate
the population mean µ is
2
 zc σ 
n= 
 E 
• If σ is unknown, you can estimate it using s,
provided you have a preliminary sample with at
least 30 members.

40
 Section 6.2 Confidence intervals for
the mean (σ unknown )
When the population standard deviation is unknown, the
random variable x is approximately normally distributed, it
follows a t-distribution.
x−µ
t=
s
Critical values of t are tc. n

d.f. = 2
d.f. = 5
t
0 41
 Confidence Intervals and
t-Distributions (σ unknown)
In Words In Symbols
1. Verify that σ is not known,
the sample is random, and
either the population is
normally distributed or
n≥30.

2. Identify the sample Σx ∑(x − x ) 2

x= s=
statistics n, x , and s. n n −1

42
Confidence Intervals and t-Distributions
In Words In Symbols
3. Identify the degrees of
freedom, the level of d.f. = n – 1
confidence c, and the
critical value tc.
s
4. Find the margin of E = tc
n
error E.
5. Find the left and right Left endpoint: x − E
endpoints and form the Right endpoint: x + E
confidence interval. Interval:
x−E<µ < x+E
43
 Normal or t-Distribution?

When both n<30 and the population is not normally

distributed, you cannot use the standard normal
distribution or the t-distribution. 44
.
 Section 6.3 Confidence interval for
population proportions
In Words In Symbols
1. Identify the sample statistics n
and x.
x
2. Find the point estimate p̂. pˆ =
n
3. Verify that the sampling
distribution of p̂ can be npˆ ≥ 5, nqˆ ≥ 5
approximated by a normal
distribution.
4. Find the critical value zc that Use the Standard
corresponds to the given level of Normal Table.
confidence c.
45
 Constructing Confidence Intervals for p
In Words In Symbols
ˆˆ
pq
5. Find the margin of error E. E = zc
n

6. Find the left and right Left endpoint: p̂ − E

endpoints and form the Right endpoint: p̂ + E
confidence interval. Interval:
pˆ − E < p < pˆ + E

46
 Sample Size
• Given a c-confidence level and a margin of error
E, the minimum sample size n needed to estimate
p is 2
 zc 
ˆ ˆ 
n = pq
E
• This formula assumes you have an estimate for p̂
and qˆ .
• If not, use pˆ = 0.5 and qˆ = 0.5.

47
 Section 6.4 Confidence intervals for
variances and standard deviation
• If the random variable x has a normal
distribution, then the distribution of
(n − 1)s 2
χ2 =
σ2
forms a chi-square distribution for samples of
any size n > 1.

48
 Find Critical Values for χ2
• Each area in the table represents the region under the
chi-square curve to the right of the critical value.

49
 Confidence Intervals for σ2 and σ
Confidence Interval for σ2:
• (n − 1)s 2 (n − 1)s 2
<σ2 <
χ R2 χ L2
Confidence Interval for σ:

• (n − 1)s 2 (n − 1)s 2
<σ <
χ R2 χ L2

• The probability that the confidence intervals contain

σ2 or σ is c.

50
 Confidence Intervals for σ2 and σ
In Words In Symbols
1. Verify that the population has a
normal distribution.
2. Identify the sample statistic n and d.f. = n – 1
the degrees of freedom.
s2 =
∑( x − x ) 2
3. Find the point estimate s2. n −1
4. Find the critical values χ2R and χ2L
that correspond to the given level Use Table 6 in
of confidence c. Appendix B.

51
 Confidence Intervals for σ2 and σ
In Words In Symbols
5. Find the left and right
(n − 1)s 2 (n − 1)s 2
endpoints and form the <σ2 <
χ R2 χ L2
confidence interval for the
population variance.
6. Find the confidence
interval for the population (n − 1)s 2 (n − 1)s 2
standard deviation by <σ <
χ R2 χ L2
taking the square root of
each endpoint.

52
 Chapter 7 Hypothesis Testing
Writing the Hypotheses
• You are given a claim about a population parameter μ,
p, σ2, or σ.
• Rewrite the claim and its complement using ≥, ≤ ,= and
<,>, ≠.
• Identify the claim. Is it H0 or Ha?
Specifying a Level of Significance
• Specify α, the maximum acceptable probability of
rejecting a valid H0 (a type I error).
Specifying the Sample Size
• Specify your sample size n.
53
Choosing the Test
Mean: H0 describes a hypothesized population mean μ.

Use a z-test when σ is known and the population is

normal.
Use a z-test for any population when σ is known and n ≥
30.
Use a t-test when σ is not known and the population is
normal.
Use a t-test for any population when σ is not known and
n ≥ 30.

54
Proportion: H0 describes a hypothesized
population proportion p.
Use a z-test for any population when np ≥ 5 and
nq ≥ 5.
Variance or Standard Deviation: H0 describes a
hypothesized population variance σ2 or standard
deviation σ.
Use a chi-square χ2 test when the population is
normal.

55
Sketching the Sampling Distribution
Use Ha to decide whether the test is left-tailed,
right-tailed, or two-tailed.
Finding the Standardized Test Statistic
• Take a random sample of size n from the
population.
• Compute the test statistic x , p̂, or s2.
• Find the standardized test statistic z, t, or χ2.

56
Making a Decision
Option 1. Decision based on rejection region
Use α to find the critical value(s) z0, t0, or χ20 and rejection region(s).
Decision Rule:
Reject H0 when the standardized test statistic is in the rejection
region.
Fail to reject H0 when the standardized test statistic is not in the
rejection region.
Option 2. Decision based on P-value
Use the standardized test statistic or technology to find the P-value.
Decision Rule:
Reject H0 when P ≤ α.
Fail to reject H0 when P > α.

57
58
59
60
 Section 8.1

Testing the Difference Between Means

(Independent Samples, σ1 and σ2 known)
Independent and Dependent Samples

Independent Samples Dependent Samples

(not related) (paired or matched)

Sample 1 Sample 2 Sample 1 Sample 2

 Two-Sample z-Test for the Difference
Between Means
If these requirements are met, the sampling distribution
for x1 − x2 (the difference of the sample means) is a
normal distribution with mean μ1 - μ2.

• Test statistic is x1 − x2.

• The standardized test statistic is

=z
(x1 − x2) − (µ1 − µ2) where σ= σ 12 σ 22
+ .
σ x −x
1 2
x −x
1 2
n1 n2
 Using a Two-Sample z-Test for the
Difference Between Means (σ1 and σ2
known)
In Words In Symbols
1. Verify σ1 and σ2 are known, the samples
are random and independent, and either
the populations are normally distributed
or both n1 ≥30 and n2 ≥30.
2. State the claim mathematically and State H0 and Ha.
verbally. Identify the null and H0: μ1 =, ≥, ≤ μ2
alternative hypotheses. Ha: μ1 ≠ ,<, >μ2
3. Specify the level of significance. Identify α.
4. Determine the critical value(s). Use Table 4 in
Appendix B.
 Using a Two-Sample z-Test for the
Difference Between Means (σ1 and σ2
In Words
known) In Symbols
5. Determine the rejection
region(s).
6. Find the standardized test
statistic and sketch the z=
( x1 − x2) − (µ1 − µ2)
σ x −x
sampling distribution. 1 2

7. Make a decision to reject or fail If z is in the rejection

to reject the null hypothesis. region, reject H0.
Otherwise, fail to
8. Interpret the decision in the reject H0.
context of the original claim.
 Section 8.2

Testing the Difference Between Means

(Independent Samples, σ1 and σ2
unknown)
Two-Sample t-Test for the Difference
Between Means
If these requirements are met, the sampling distribution
for x1 − x2 (the difference of the sample means) is
approximated by a t-distribution with mean μ1 - μ2.
• The test statistic is x1 − x2 .

• The standardized test statistic is

t=
( x1 − x2) − (µ1 − µ2)
.
sx − x
1 2
 Two-Sample t-Test for the Difference Between
Means
• Variances are equal
– Information from the two samples is combined to
calculate a pooled estimate of the standard deviation
σˆ.
σˆ =
(n1 − 1) s12 + (n2 − 1) s22
n1 + n2 − 2
 The standard error for the sampling distribution of
x1 − x2 is
1 1
sx − x =σˆ ⋅ +
1 2
n1 n2
 d.f.= n1 + n2 – 2
 Two-Sample t-Test for the Difference
Between Means
• Variances are not equal
– If the population variances are not equal, then the
standard error is
2 2
s1 s2
sx= + .
1
−x 2
n1 n2
– d.f.= smaller of n1 – 1 and n2 – 1
 Normal or t-Distribution?
Are both population Are both populations
Yes normal or both sample Yes Use the z-test.
standard deviations
known? sizes at least 30?
No No

Are both populations normal or You cannot use the z-test or the
No
both sample sizes at least 30? t-test.
Use the t-test with
Yes 1 1
s x − x σˆ
= +
n1 n2
Are the population variances equal?
1 2

d.f = n1 + n2 – 2.
Use the t-test with
s12 s22
sx − x = +
1 2
n1 n2
d.f = smaller of n1 – 1 or n2 – 1.
 Two-Sample t-Test for the Difference
Between Means
(Independent Samples, σ1 and σ2 unknown)
In Words In Symbols
1. Verify σ1 and σ2 are unknown, the
samples are random and
independent, and either the
populations are normally distributed
or both n1 ≥30 and n2 ≥30.
2. State the claim mathematically and State H0 and Ha.
verbally. Identify the null and H0: μ1 =, ≥, ≤ μ2
alternative hypotheses. Ha: μ1 ≠ ,<, >μ2

3. Specify the level of significance. Identify α.

4. Determine the degrees of freedom. d.f. = n1+ n2 – 2 or
d.f. = smaller of n1 – 1 or n2 – 1.
 Two-Sample t-Test for the Difference
Between Means
(Independent Samples, σ1 and σ2 unknown)
In Words In Symbols
5. Determine the critical value(s). Use Table 5 in Appendix
B.
6. Determine the rejection region(s).
7. Find the standardized test statistic
t=
(x1 − x2)− (µ1 − µ2)
and sketch the sampling distribution. sx − x
1 2

8. Make a decision to reject or fail to If t is in the rejection

reject the null hypothesis. region, reject H0.
Otherwise, fail to reject
9. Interpret the decision in the context H0.
of the original claim.
 Section 8.3

Testing the Difference Between

Means (Dependent Samples)
 t-Test for the Difference Between
Means
A t-test can be used to test the mean of the differences
for a population of paired data when (1) the samples are
random, (2) the samples are dependent (paired), (3) the
population are normally distributed or n ≥30.

•The test statistic is

∑d
d= .
n
•The standardized test statistic is
d − µd
t= .
•The degrees of freedom are d.f. = n – 1. sd n
 t-Test for the Difference Between
Means (Dependent Samples)
In Words In Symbols
1. Verify the samples are random
and dependent, and either the
populations are normally
distributed or n ≥30.
2. State the claim mathematically State H0 and Ha.
and verbally. Identify the null H0: μd =, ≥, ≤ 0
and alternative hypotheses. Ha: μd ≠ ,<, >0

3. Specify the level of Identify α.

significance.
 t-Test for the Difference Between
Means (Dependent Samples)
In Words In Symbols
4. Determine the degrees of d.f. = n – 1
freedom.
5. Determine the critical Use Table 5 in
value(s). Appendix B.
6. Determine the rejection
region(s).
7. Calculate d and sd. d = ∑d
n 2
2 (Σd )
∑(d − d ) 2 Σd −
=sd = n
n −1 n −1
 t-Test for the Difference Between
Means (Dependent Samples)
In Words In Symbols
8. Find the standardized test d − µd
statistic. t=
sd n
9. Make a decision to reject or If t is in the rejection
fail to reject the null region, reject H0.
hypothesis. Otherwise, fail to
reject H0.
10. Interpret the decision in the
context of the original
claim.
 Section 8.4

Testing the Difference Between

Proportions
Two-Sample z-Test for the Difference
Between Proportions

x1 + x2
p= and q = 1 − p.
n1 + n2
Two-Sample z-Test for the Difference
Between Proportions
In Words In Symbols

x1 + x2
p=
n1 + n2

State H0 and Ha.

H0: p1 =, ≥, ≤ p2
Ha: p1 ≠ ,<, > p2
Identify α.
Use Table 4 in
Appendix B.
 Two-Sample z-Test for the Difference
Between Proportions
In Words In Symbols
7. Find the standardized test
statistic and sketch the
sampling distribution.

8. Make a decision to reject or If z is in the rejection

fail to reject the null region, reject H0.
hypothesis. Otherwise, fail to
reject H0.
9. Interpret the decision in the
context of the original claim.
 9.1 Correlation
Correlation coefficient (Pearson product
moment correlation coefficient)
• A measure of the strength and the direction of a linear
relationship between two variables.
• The symbol r represents the sample correlation
coefficient.
• A formula for r is
n ∑ xy − (∑ x)(∑ y) n is the number
r= of data pairs
2 2
n ∑ x 2 − (∑ x) n ∑ y 2 − (∑ y)
• The population correlation coefficient is represented
.
by ρ (rho). 82
 The t-Test for the Correlation
Coefficient
• A t-test can be used to test whether the correlation
between two variables is significant.
• The test statistic is r
• The standardized test statistic
r r
t
= =
σr 1− r2
n−2
follows a t-distribution with d.f. = n – 2.
• In this text, only two-tailed hypothesis tests for ρ
are considered.
H0: ρ = 0 (no significant correlation)
. Ha: ρ ≠ 0 (significant correlation) 83
 Using the t-Test for ρ
In Words In Symbols
1. State the null and alternative State H0 and Ha.
hypothesis.
2. Specify the level of Identify α.
significance.
3. Identify the degrees of
d.f. = n – 2.
freedom.
4. Determine the critical
value(s) and rejection Use Table 5 in
region(s). Appendix B.

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 Using the t-Test for ρ
In Words In Symbols
r
5. Find the standardized test t=
1− r2
statistic. n−2

6. Make a decision to reject or If t is in the rejection

fail to reject the null region, reject H0.
hypothesis. Otherwise fail to reject
H0.

7. Interpret the decision in the

context of the original claim.

. 85
 9.2 The Equation of a Regression Line
• ŷ = mx + b where
n ∑ xy − (∑ x)(∑ y) ∑y ∑x
m= b =y − mx = − m
n ∑ x 2 − (∑ x)
2
n n

• y is the mean of the y-values in the data

• x is the mean of the x-values in the data
• The regression line always passes through the
point ( x , y )

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