Math 115 - Elementary Statistics Summary*

CHAPTER 1
Section 1.1
Data Sets
Population (parameter is numerical characteristic)
Sample (statistic is numerical characteristic)
Branches of Statistics
Descriptive and Inferential

Section 1.2
Types of Data
Qualitative and Quantitative
Levels of Measure
nominal, ordinal, interval, and ratio

Section 1.3
Data Collection Methods
1. Observational Study 3. Simulation
2. Experiment 4. Survey

Types of Sampling Techniques

1. Random sample
2. Stratified sample
3. Cluster sample
4. Systematic sample
5. Convenience sample

CHAPTER 2
Section 2.1
Frequency Distribution Columns
Class, Class Boundaries, Frequency, Midpoint, Relative Frequency, Cumulative
Frequency.

range
Class Width =
# of classes

lower limit+upper limit

Midpoint =
2

class frequency
Relative Frequency =
sample size(n)

Frequency Histogram (horizontal = midpoints, vertical = frequencies)

Section 2.3
∑X ∑X
Pop. Mean: µ = ̅
Sample Mean: X = Weighted Mean: x̅ =
∑(x •w)
N n ∑w

1
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*

Mean of Grouped Data (mean of a frequency distribution)

∑(x •f)
x̅ = x = midpoints, f = frequencies, n = ∑f
n

Section 2.4
Population deviation of x = x - µ Sample deviation of x = x - 𝑋̅
Sum of Squares: ∑( x - µ)2

Population Standard Deviation: Sample Standard Deviation:

∑(x− µ)2 ∑(x− x̅)2
σ=√ s=√
N n−1

Calculator: Computing Standard Deviation

To enter the data list into the calculator:
STAT → EDIT Menu → enter data into L1
To compute mean and standard deviation
STAT → CALC Menu → 1:1 Var Stats

Empirical Rule

Chebychev’s Theorem
The portion of any data set lying within K (K > 1) standard deviations from the mean is
1
at least 1- 2
k
If K = 2 then at least 75% of data lies within 2 standard deviation of the mean.
If K = 3 then at least 88.9% of data lies within 3 standard deviations of the mean.

2
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*
Standard Deviation of Grouped Data (s.d. of a frequency distribution)
∑(x− x̅)2 •𝑓
S=√
n−1
Calculator: L1 = midpoints (x-values), L2= frequencies; then use 1-var stats then L1, L2

Section 2.5
IQR = Q3 – Q1
Outlier: any entry beyond: Q1 – 1.5(IQR) or Q3 + 1.5(IQR)

# of data values less than x

Percentile of x = • 100
total number of data values
x− µ
z-score = (A z-score is considered unusual if it is outside of the -2 to 2 range)
σ

CHAPTER 3
Section 3.1
Fundamental Counting Principle: multiple events occurring in sequence m•n ways

Classical (Theoretical) Probability Empirical Probability

# of outcomes in event E frequency of event f
P(E) = P(E) = =
# of outcomes in sample space total frequency n

Compliment: P(E)’ = 1 – P(E)

Section 3.2
Independent Events: P(B/A) = P(B) and P(A/B) = P(A)

Multiplication Rule (probability that two events will occur in sequence)

P(A and B) = P(A) • P(B/A) independent events: P(A andB) = P(A) • P(B)

Section 3.3
Addition Rule
P(A or B) = P(A) + P(B) – P(A and B) mutually exclusive: P(A or B) = P(A) + P(B)

CHAPTER 4
Section 4.1
Mean of a Discrete Probability Distribution: µ = ∑ x • p(x)

Standard Deviation of a Discrete Probability Distribution (Discr. Random Variable)

σ = √∑(𝑥 − µ)2 • 𝑝(𝑥)

Calculator for Standard Deviation of Discrete Probability Distribution:

L1 – discrete random variables (x); L2 – probabilities p(x); then 1-Var stats then L1, L2

3
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*

Expected Value: E(x) = µ = ∑ x • p(x)

Section 4.2
Binomial Experiments
n = number of trials; p = p(success); q = p(failure); x = # of successes in n trials
n!
Binomial Probability Formula: • px • qn-x p(exactly x successes in n trials)
(n−x)! •x!

Calculator for Binomial Probabilities:

Probability of exactly x success: binompdf(n, p, x)
Probability of “at most x successes” binomcdf(n, p, x)

Unusual Probabilities: p ≤ .05

Population Parameters of a Binomial Distribution

Mean: µ = n•p
Variance: σ2 = n•p•q
Standard Deviation: σ = √n • p • q

CHAPTER 5
Section 5.1
To transform any x-value to a z-score use:
x− µ value−mean
z-score = =
σ standard deviation

Calculator to find an area that corresponds to a given z-score:

normalcdf(-10,000,z) = area to the left of z
normalcdf(z, 10,000) = area to the right of z
normalcdf(z1, z2) = area between two z’s

Section 5.2
Finding Normal Distribution Probabilities
Finding the probability that x will fall in a given interval by finding the area under the
normal curve for that interval
Calculator: normalcdf(x1, x2, µ, σ) (Probability from raw data (x’s))

Section 5.3
Calculator to find the z-score for a given area or a percentile:
invNorm(area)

Finding an x-value for a corresponding z-score

x = µ + zσ

Calculator to find an x-value for a given probability:

Calculator: invNorm(area, µ, σ)

4
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*

Section 5.4
Central Limit Theorem
If n ≥ 30 or population is normally distributed, then:
𝜎2 𝜎
𝜇𝑥̅ = 𝜇 and 𝜎𝑥̅2 = and 𝜎𝑥̅ =
𝑛 √𝑛
To transform 𝐱̅ to a z-score:
x̅− µx̅
Z= σ
√n
𝜎
Calculator: normalcdf (x1, x2, 𝜇𝑥̅ , )
√𝑛

Section 5.5
You can use a normal distribution to approximate a binomial distribution if np ≥ 5 and
nq ≥ 5. If this is true, then do the following:
1. Find µ = np and σ = √𝑛𝑝𝑞
2. Apply the continuity correction (Add or subtract 0.5 from the endpoints).
3. Use the calculator to find the binomial probability:
normalcdf: (x1, x2, µ, σ)

CHAPTER 6
Section 6.1 (Confidence interval for the mean - large samples)
Margin of Error (E): The greatest possible distance between x̅ and µ
𝜎
E = zc
√𝑛

Confidence Interval: where “c” is the probability that the confidence interval contains µ
x̅ – E < µ < x̅ + E

Calculator: STAT → TESTS Menu → 7:Zinterval

Minimum Sample Size:

zc 𝜎 2
n=( )
E

Section 6.2 (Confidence interval for the mean - small samples)

Use when: σ is unknown, n < 30 and population is (approx.) normally distributed
Degrees of Freedom:
d.f. = n – 1

Critical Value = tc is found in Table 5 using d.f. and the confidence interval wanted.

Margin of Error (E):

𝑠
E = tc
√𝑛
5
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*

Confidence Interval: x̅ – E < µ < x̅ + E

Calculator: STAT → TESTS Menu → 8:Tinterval

Section 6.3 (Confidence intervals for population proportions)

Population Proportion (p):
- probability of success in a single trial of a binomial experiment
- proportion of the population included in a “success” outcome (we are estimating this)

x # of successes in the sample

p̂ = =
n sample size

̂ =1-𝐩
𝐪 ̂

Confidence Interval for p: p

̂ - E < p < p̂ + E

Margin of Error ( E):

p̂ q
̂
E = zc√ (np̂ ≥ 5 and nq̂ ≥ 5 for a normal approximation)
n

Calculator: STAT → TESTS Menu → A:1-PropZint

Minimum Sample Size:

zc 2
n=p
̂ q̂ ( )
E

CHAPTER 7
Section 7.1
Hypothesis Testing: Uses sample statistics to test a claim about the value of a
population parameter.

H0: μ ≥ k H0: μ ≤ k H0: μ = k

Ha: μ < k Ha: μ > k Ha: μ ≠ k
left-tailed right-tailed two-tailed

Level of Significance = 𝛂
The maximum allowable probability of making a Type I error.
P-Value (probability value)
-The estimated probability of rejecting Ho when it is true (Type I error)
-The smaller the P-value the more evidence to reject Ho.

6
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*

Section 7.2 (Hypothesis testing for mean - large sample)

z-Test
x̅− µ x̅− µ
z = σ = s (if n ≥ 30, the σ ≈ s)
√n √n

Guidelines for Using P-Values

1. Find the z-score and then area of your data and compare it to α.
can use normalcdf( ∞, x, ̅ µx̅, σx̅ ) = area of data
2. If P ≤ α then reject Ho.
If P > α then fail to reject Ho.

Calculator: STAT → TESTS Menu → 1:Z-Test

Rejection Regions
-Range of values for which Ho is not probable; If z-score for data is in this region
reject Ho.

Guidelines for Using Rejection Regions

1. Find the z-score that goes with α and sketch. (This delineates rejection region)
2. Find z-score for given data and add to sketch
3. Reject Ho if data z-score is in rejection region.

Section 7.3 Hypothesis Testing for the mean - small samples using t-Distribution)

Using t-Test Guidelines

1. Find critical values (t-scores) for α using d.f. = n – 1, and table 5 then sketch
2. Compute t for data and add to sketch
3. Reject Ho if t for data is in rejection region delineated by critical values.

x̅− µ
t= s
√n

Using P-Values with t-Test

This can be done only with a graphing calculator
Calculator: STAT → TESTS Menu → 2:T-Test

7
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*
Section 7.4 (Hypothesis testing for a population proportion (p))

Test statistic = 𝑝̂ and standardized test statistic = z

Must have: np≥5 and nq≥5 then use z-Test:
̂−p
p
Z=
pq
√
n

Guidelines for Hypothesis Testing For a Population Proportion

1. check np and nq then find rejection regions for α and sketch
2. Find z-scores for data and add to sketch
3. Reject Ho if data z-score is in rejection region.

Calculator: STAT → TESTS Menu → 5:PropZTest

CHAPTER 8
Section 8.1 (Testing the difference between sample means - large sample)
Necessary z-Test Conditions
1. Samples are randomly selected
2. Samples are independent
3. n≥30 or each population is normally distributed and σ is known.

Then x̅1 - x̅2 is normally distributed so you can use a z-Test

(s1 and s2 can be used for σ1 and σ2)

(x̅1 − x̅ 2 )− (µ1 − µ2 )
z=
σ 2 σ 2
√ 1 + 2
n1 n2

Calculator: STAT → TESTS Menu → 3:2-SampZTest

Section 8.2 (Testing the difference between sample means - small sample)
-n<30 and σ is unknown
-Samples must be independent, randomly selected and normally distributed

If the variances are equal use the following to compute t (pooled estimate):

(x̅1 − x̅2 )− (µ1 − µ2 )

t= and d.f. = n1 + n2 − 2
(n −1 )s1 2 + (n2 −1)s2 2 1 1
√ 1 •√ +
n1 + n2 −2 n1 n2

8
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*
If the variances are not equal use the following to compute t :

(x̅1 − x̅2 )− (µ1 − µ2 )

t= and d.f. = smaller of (n1 – 1) and (n2 – 1)
s1 2 s2 2
√ +
n1 n2

Calculator: STAT → TESTS Menu → 4:2-SampTTest Pooled: Yes or no

Section 8.4 (Testing the difference between population proportions)

To use a z-Test
1. The samples are independent and randomly selected.
2. n1p1, n1q1, n2p2, n2q2 all ≥ 5 (large enough to use a normal sampling distribution)

Weighted Estimate of p1 and p2

x1 + x2
p̅ = x1 = n1p
̂1 and x2 = n2p
̂2 (assume that p2 – p1 = 0)
n1 + n2

q̅ = 1 - p̅ (Condition needed: n1𝑝

̅̅̅,
1 n1̅̅̅,
𝑞1 n2̅̅̅,
𝑝2 n2̅̅̅
𝑞2 all ≥ 5)

̂1 − p
(p ̂2 ) – (p1 − p2 )
Z=
1 1
̅q
√p ̅( + )
n n 1 2

Calculator: STAT → TESTS Menu → 6:2-PropZTest

CHAPTER 9
Section 9.1
Correlation Coefficient (r)
-measures the direction and strength of a linear correlation between two variables
-range: -1 ≤ r ≤ 1

Correlation Coefficient Formula

n∑xy – (∑x)(∑y)
r=
√𝑛∑𝑥 2 − (∑𝑥)2 √𝑛∑𝑦 2 − (∑𝑦)2

Calculator:
STAT → Edit → L1 (enter x-values) and L2 (enter y-values), then
STAT → CALC Menu → 4: LinReg (ax + b) → enter

Testing a Population Correlation Coefficient With Table 11

1. Determine n = # of pairs.
2. Find the critical values for α using Table 11.
3. If |r| > c.v. the correlation coefficient of the population can be determined to be
significant.

9
*Reproduced with permission from Stacey Buck
 Math 115 - Elementary Statistics Summary*
Hypothesis Testing for a Population Correlation Coefficient 𝝆
Ho: 𝜌 = 0 (no significant correlation)
Ha: 𝜌 ≠ 0 (significant correlation)

r
t= d.f = n – 2
2
√1− r
n−2

Section 9.2
Equation of a Regression Line:
ŷ = mx + b

CHAPTER 10
Section 10.1
Chi-Square Goodness-of-fit Test: Used to test whether a frequency distribution fits
an expected distribution.
Ho: The frequency distribution fits the specified distribution
Ha: The frequency distribution does not fit the specified distribution.
Ei = npi
n = the number of trials (sample size)
pi = the assumed probability of the specific category.
Conditions Needed:
1. The observed frequencies must be obtained using a random sample
2. Each E ≥ 5

(O−E)2
x2 = ∑ d.f. = k – 1 (k = # of categories in the distribution)
E

Guidelines For Performing a Chi-Square Goodness-o-Fit Test

1. Use d.f. and Table 6 to find the critical values and sketch the rejection region
2. Compute x2 and add to sketch.
3. If x2 is in rejection region reject Ho.

Section 10.2
Chi-Square Independence Test: Used to determine whether the occurrence of one
variable affects the probability of the occurrences of the other variable.
(O−E)2
x2 = ∑ d.f. = (r - 1)(c - 1) (r = # of rows and c = # of columns)
E

Guidelines For Performing a Chi-Square Independence Test

1. Use d.f. and Table 6 to find the critical values and sketch the rejection region
2. Compute x2 and add to sketch.
3. If x2 is in rejection region reject Ho.

10
*Reproduced with permission from Stacey Buck