Review
• Variables - Catigorical v.s. Quantitative
• Variables - Catigorical v.s. Quantitative
• Graphs for distributional information: Pie chart, Bar graph,
Histogram, Stemplot, Timeplot, Boxplot
• Variables - Catigorical v.s. Quantitative
• Graphs for distributional information: Pie chart, Bar graph,
Histogram, Stemplot, Timeplot, Boxplot
• Overall pattern of the graph: Symetric/Skewed, Center,
Spread, Outlier, Trend
• Measure of center: Mean/Median
• Measure of center: Mean/Median
• Measure of variability: Quartiles (Q1 , Q2 , Q3 ), Range, IQR,
1.5×IQR rule, Outlier, Variance, Standard deviation
• Measure of center: Mean/Median
• Measure of variability: Quartiles (Q1 , Q2 , Q3 ), Range, IQR,
1.5×IQR rule, Outlier, Variance, Standard deviation
• Five-number summary, Boxplot
• Density curve
• Density curve
• Normal distributions / Normal curves
• Density curve
• Normal distributions / Normal curves
• z-score, Standard normal distribution
• Density curve
• Normal distributions / Normal curves
• z-score, Standard normal distribution
• 68 − 95 − 99.7 rule, Probabilities for normal distribution
• Explanatory variable / Response variable
• Explanatory variable / Response variable
• Scatterplot: Direction (Positive / Negative), Form (Linear /
Nonlinear), Strength, Outlier
• Explanatory variable / Response variable
• Scatterplot: Direction (Positive / Negative), Form (Linear /
Nonlinear), Strength, Outlier
• Correlation
• Linear regression: ŷ = a + bx; Slope b, Intercept a,
Predication
• Linear regression: ŷ = a + bx; Slope b, Intercept a,
Predication
• Correlation and regression, r 2 , Residual
• Linear regression: ŷ = a + bx; Slope b, Intercept a,
Predication
• Correlation and regression, r 2 , Residual
• Cautions for regression: Influential observations,
Extrapolation, Lurking variables
• Sample / Population
• Sample / Population
• Random sampling design: Simple random sample (SRS),
Stratified random sample, Multistage sample
• Sample / Population
• Random sampling design: Simple random sample (SRS),
Stratified random sample, Multistage sample
• Bad samples: Voluntary response sample, Convenience sample
• Observational studies & Experimental studies (experiments)
• Observational studies & Experimental studies (experiments)
• Treatments / Factors
• Observational studies & Experimental studies (experiments)
• Treatments / Factors
• Design of experiments:
• Observational studies & Experimental studies (experiments)
• Treatments / Factors
• Design of experiments:
control (comparison, placebo)
• Observational studies & Experimental studies (experiments)
• Treatments / Factors
• Design of experiments:
control (comparison, placebo)
randomization (table of random digits, double-blind)
• Observational studies & Experimental studies (experiments)
• Treatments / Factors
• Design of experiments:
control (comparison, placebo)
randomization (table of random digits, double-blind)
matched pairs design / Block design
• Probability: Sample space (S) & Events
• Probability: Sample space (S) & Events
• Rules for probability model:
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
2. for sample space S, P(S) = 1
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
2. for sample space S, P(S) = 1
3. if two events A and B are disjoint, then
P(A or B) = P(A) + P(B)
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
2. for sample space S, P(S) = 1
3. if two events A and B are disjoint, then
P(A or B) = P(A) + P(B)
4. for any event A, P(A does not occur) = 1 − P(A)
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
2. for sample space S, P(S) = 1
3. if two events A and B are disjoint, then
P(A or B) = P(A) + P(B)
4. for any event A, P(A does not occur) = 1 − P(A)
• Discrete probability models / Continuous probability models
• Probability: Sample space (S) & Events
• Rules for probability model:
1. for any event A, 0 ≤ P(A) ≤ 1
2. for sample space S, P(S) = 1
3. if two events A and B are disjoint, then
P(A or B) = P(A) + P(B)
4. for any event A, P(A does not occur) = 1 − P(A)
• Discrete probability models / Continuous probability models
• Random variables / Distributions
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Sampling distribution of the sample mean x̄ for an SRS:
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Sampling distribution of the sample mean x̄ for an SRS:
∗ mean of x̄ equals the population mean µ
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Sampling distribution of the sample mean x̄ for an SRS:
∗ mean of x̄ equals the population mean µ
∗ standard deviation of x̄ equals √σn , where σ is the
population standard deviation and n is the sample size
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Sampling distribution of the sample mean x̄ for an SRS:
∗ mean of x̄ equals the population mean µ
∗ standard deviation of x̄ equals √σn , where σ is the
population standard deviation and n is the sample size
∗ if the population has a normal distribution, then
√
x̄ ∼ N(µ, σ/ n)
• Population / Sample; Parameters / Statistics
µ / x̄, σ / s, p / p̂
• Statistics are random variables
• Sampling distribution of the sample mean x̄ for an SRS:
∗ mean of x̄ equals the population mean µ
∗ standard deviation of x̄ equals √σn , where σ is the
population standard deviation and n is the sample size
∗ if the population has a normal distribution, then
√
x̄ ∼ N(µ, σ/ n)
∗ central limit theorem: if the sample size is large
(n ≥ 30), then x̄ is approximately normal, i.e.
√
approx
x̄ ∼ N(µ, σ/ n)
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
∗ form: estimate ± margin of error / interpretation
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
∗ form: estimate ± marginof error / interpretation
σ
σ
∗ x̄ − z ∗ √ , x̄ + z ∗ √
n
n
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
∗ form: estimate ± marginof error / interpretation
σ
σ
∗ x̄ − z ∗ √ , x̄ + z ∗ √
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
∗ P-value:
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to z
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to z
? Ha : µ < µ0 — lower tail probability corresponding
to z
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to z
? Ha : µ < µ0 — lower tail probability corresponding
to z
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |z|
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: z =
σ/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to z
? Ha : µ < µ0 — lower tail probability corresponding
to z
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |z|
∗ significance level α and conclusion
• Assumptions for z-procedures:
• Assumptions for z-procedures:
∗ the sample is an SRS
• Assumptions for z-procedures:
∗ the sample is an SRS
∗ the population has a normal distribution
• Assumptions for z-procedures:
∗ the sample is an SRS
∗ the population has a normal distribution
∗ the population standard deviation σ is known
• Assumptions for z-procedures:
∗ the sample is an SRS
∗ the population has a normal distribution
∗ the population standard deviation σ is known
• Margin of errors in confidence intervals are affected by C , σ
and n
to get a level C C.I. with margin of m, we need an SRS
with sample size
∗ 2
z σ
n=
m
• Assumptions for z-procedures:
∗ the sample is an SRS
∗ the population has a normal distribution
∗ the population standard deviation σ is known
• Margin of errors in confidence intervals are affected by C , σ
and n
to get a level C C.I. with margin of m, we need an SRS
with sample size
∗ 2
z σ
n=
m
• The significance of test will also be affected by sample size
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
s
• Standard error: √
n
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
s
• Standard error: √
n
• t-distribution; degrees of freedom (n − 1)
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
s
• Standard error: √
n
• t-distribution; degrees of freedom (n − 1)
• Confidence intervals:
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
s
• Standard error: √
n
• t-distribution; degrees of freedom (n − 1)
• Confidence
intervals:
∗ s
∗ s
∗ x̄ − t √ , x̄ + t √
n
n
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
s
• Standard error: √
n
• t-distribution; degrees of freedom (n − 1)
• Confidence
intervals:
∗ s
∗ s
∗ x̄ − t √ , x̄ + t √
n
n
∗ t ∗ is determined by the confidence level C — the t-score
corresponding to the upper tail (1 − C )/2
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
∗ P-value:
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to t
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |t|
• Inference about µ with unknown σ — t-procedures
(confidence interval & test of significance)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ = µ0
x̄ − µ0
√
∗ test statistics: t =
s/ n
∗ P-value:
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |t|
∗ significance level α and conclusion
• Inference about two means — µ1 − µ2
• Inference about two means — µ1 − µ2
• Standard error for x̄1 − x̄2 :
s
s12
s2
+ 2
n1 n2
• Inference about two means — µ1 − µ2
• Standard error for x̄1 − x̄2 :
s
s12
s2
+ 2
n1 n2
• Confidence interval for µ1 − µ2 :
• Inference about two means — µ1 − µ2
• Standard error for x̄1 − x̄2 :
s
s12
s2
+ 2
n1 n2
• Confidence interval for µ1 − µ2 :
s
s
2
2
s1
s2
s12
s22
∗
∗
∗ (x̄1 − x̄2 ) − t
+ , (x̄1 − x̄2 ) + t
+
n1 n2
n1 n2
• Inference about two means — µ1 − µ2
• Standard error for x̄1 − x̄2 :
s
s12
s2
+ 2
n1 n2
• Confidence interval for µ1 − µ2 :
s
s
2
2
s1
s2
s12
s22
∗
∗
∗ (x̄1 − x̄2 ) − t
+ , (x̄1 − x̄2 ) + t
+
n1 n2
n1 n2
∗ t ∗ is determined by the confidence level C — the t-score
corresponding to the upper tail (1 − C )/2
• Inference about two means — µ1 − µ2
• Standard error for x̄1 − x̄2 :
s
s12
s2
+ 2
n1 n2
• Confidence interval for µ1 − µ2 :
s
s
2
2
s1
s2
s12
s22
∗
∗
∗ (x̄1 − x̄2 ) − t
+ , (x̄1 − x̄2 ) + t
+
n1 n2
n1 n2
∗ t ∗ is determined by the confidence level C — the t-score
corresponding to the upper tail (1 − C )/2
∗ degrees of freedom: smaller of n1 − 1 and n2 − 1
• Inference about two means — µ1 − µ2
• Inference about two means — µ1 − µ2
• Test of significance:
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
? degrees of freedom: smaller of n1 − 1 and n2 − 1
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
? degrees of freedom: smaller of n1 − 1 and n2 − 1
? Ha : µ > µ0 — upper tail probability corresponding
to t
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
? degrees of freedom: smaller of n1 − 1 and n2 − 1
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
? degrees of freedom: smaller of n1 − 1 and n2 − 1
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |t|
• Inference about two means — µ1 − µ2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : µ1 = µ2 (µ1 − µ2 = 0)
x̄1 − x̄2
∗ test statistics: t = q 2
s22
s1
n1 + n2
∗ P-value:
? degrees of freedom: smaller of n1 − 1 and n2 − 1
? Ha : µ > µ0 — upper tail probability corresponding
to t
? Ha : µ < µ0 — lower tail probability corresponding
to t
? Ha : µ 6= µ0 — twice upper tail probability
corresponding to |t|
∗ significance level α and conclusion
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Sampling distribution of the sample proportion p̂ for an SRS:
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Sampling distribution of the sample proportion p̂ for an SRS:
∗ mean of p̂ equals the population proportion p
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Sampling distribution of the sample proportion p̂ for an SRS:
∗ mean of p̂ equals the population
rproportion p
p(1 − p)
∗ standard deviation of p̂ equals
n
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Sampling distribution of the sample proportion p̂ for an SRS:
∗ mean of p̂ equals the population
rproportion p
p(1 − p)
∗ standard deviation of p̂ equals
n
∗ If the sampler
size is large, p̂ is approximately normal, i.e.
p(1 − p)
approx
p̂ ∼ N(p,
)
n
• Inference about population proportion p — z-procedures
(confidence interval & test of significance)
• Sampling distribution of the sample proportion p̂ for an SRS:
∗ mean of p̂ equals the population
rproportion p
p(1 − p)
∗ standard deviation of p̂ equals
n
∗ If the sampler
size is large, p̂ is approximately normal, i.e.
p(1 − p)
approx
p̂ ∼ N(p,
)
rn
p̂(1 − p̂)
• Standard error of p̂:
n
• Inference about population proportion p — z-procedures
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 15 and n(1 − p̂) ≥ 15
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 15 and n(1 − p̂) ≥ 15
• Plus four confidence intervals:
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 15 and n(1 − p̂) ≥ 15
• Plus four confidence
intervals: r
r
p̃(1
−
p̃)
p̃(1 − p̃)
∗
∗
∗ p̃ − z
, p̃ + z
n+4
n+4
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 15 and n(1 − p̂) ≥ 15
• Plus four confidence
intervals: r
r
p̃(1
−
p̃)
p̃(1 − p̃)
∗
∗
∗ p̃ − z
, p̃ + z
n+4
n+4
number of successes in the sample + 2
∗ p̃ =
n+4
• Inference about population proportion p — z-procedures
• Large-sample confidence intervals:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
∗
∗
, p̂ + z
∗ p̂ − z
n
n
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 15 and n(1 − p̂) ≥ 15
• Plus four confidence
intervals: r
r
p̃(1
−
p̃)
p̃(1 − p̃)
∗
∗
∗ p̃ − z
, p̃ + z
n+4
n+4
number of successes in the sample + 2
∗ p̃ =
n+4
∗ Use it when the confidence level is at least 90% and the
sample size n is at least 10
• Inference about population proportion p — z-procedures
• Inference about population proportion p — z-procedures
• Test of significance:
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
? Ha : p > p0 — upper tail probability corresponding
to z
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
? Ha : p > p0 — upper tail probability corresponding
to z
? Ha : p < p0 — lower tail probability corresponding
to z
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
? Ha : p > p0 — upper tail probability corresponding
to z
? Ha : p < p0 — lower tail probability corresponding
to z
? Ha : p 6= p0 — twice upper tail probability
corresponding to |z|
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
? Ha : p > p0 — upper tail probability corresponding
to z
? Ha : p < p0 — lower tail probability corresponding
to z
? Ha : p 6= p0 — twice upper tail probability
corresponding to |z|
∗ significance level α and conclusion
• Inference about population proportion p — z-procedures
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p = p0
p̂ − p0
∗ test statistics: z = q
p0 (1−p0 )
n
∗ P-value:
? Ha : p > p0 — upper tail probability corresponding
to z
? Ha : p < p0 — lower tail probability corresponding
to z
? Ha : p 6= p0 — twice upper tail probability
corresponding to |z|
∗ significance level α and conclusion
∗ use this test when np0 ≥ 10 and n(1 − p0 ) ≥ 10
• Inference about two proportions — p1 − p2
• Inference about two proportions — p1 − p2
• Sampling distribution of p̂1 − p̂2 :
• Inference about two proportions — p1 − p2
• Sampling distribution of p̂1 − p̂2 :
∗ mean of p̂1 − p̂2 is p1 − p2
• Inference about two proportions — p1 − p2
• Sampling distribution of p̂1 − p̂2 :
∗ mean of p̂1 − p̂2 is p1 − p2
∗ standard
deviation of p̂1 − p̂2 is
s
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
• Inference about two proportions — p1 − p2
• Sampling distribution of p̂1 − p̂2 :
∗ mean of p̂1 − p̂2 is p1 − p2
∗ standard
deviation of p̂1 − p̂2 is
s
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
∗ If the sample size is large, p̂1 − p̂2 is approximately normal
• Inference about two proportions — p1 − p2
• Sampling distribution of p̂1 − p̂2 :
∗ mean of p̂1 − p̂2 is p1 − p2
∗ standard
deviation of p̂1 − p̂2 is
s
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
∗ If the sample size is large, p̂1 − p̂2 is approximately normal
s
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
• Standard error of p̂:
+
n1
n2
• Inference about two proportions — p1 − p2
• Inference about two proportions — p1 − p2
• Large-sample confidence intervals:
• Inference about two proportions — p1 − p2
• Large-sample confidence intervals:
∗
∗
∗ (p̂1 − p̂2 ) − z SE, (p̂1 + p̂2 ) + z SE , where SE is the
standard error of p̂1 − p̂2 :
s
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
SE =
+
n1
n2
• Inference about two proportions — p1 − p2
• Large-sample confidence intervals:
∗
∗
∗ (p̂1 − p̂2 ) − z SE, (p̂1 + p̂2 ) + z SE , where SE is the
standard error of p̂1 − p̂2 :
s
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
SE =
+
n1
n2
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
• Inference about two proportions — p1 − p2
• Large-sample confidence intervals:
∗
∗
∗ (p̂1 − p̂2 ) − z SE, (p̂1 + p̂2 ) + z SE , where SE is the
standard error of p̂1 − p̂2 :
s
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
SE =
+
n1
n2
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 10 and n(1 − p̂) ≥ 10
• Inference about two proportions — p1 − p2
• Large-sample confidence intervals:
∗
∗
∗ (p̂1 − p̂2 ) − z SE, (p̂1 + p̂2 ) + z SE , where SE is the
standard error of p̂1 − p̂2 :
s
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
SE =
+
n1
n2
∗ z ∗ is determined by the confidence level C — the z-score
corresponding to the upper tail (1 − C )/2
∗ Use it only when np̂ ≥ 10 and n(1 − p̂) ≥ 10
• Inference about two proportions — p1 − p2
• Inference about two proportions — p1 − p2
• Plus four confidence intervals:
• Inference about two proportions — p1 − p2
• Plus four confidence intervals:
∗
∗
∗ (p̃1 − p̃2 ) − z SE, (p̃1 + p̃2 ) + z SE , where SE is the
standard error of p̃1 − p̃2 :
s
p̃1 (1 − p̃1 ) p̃2 (1 − p̃2 )
+
SE =
n1 + 2
n2 + 2
• Inference about two proportions — p1 − p2
• Plus four confidence intervals:
∗
∗
∗ (p̃1 − p̃2 ) − z SE, (p̃1 + p̃2 ) + z SE , where SE is the
standard error of p̃1 − p̃2 :
s
p̃1 (1 − p̃1 ) p̃2 (1 − p̃2 )
+
SE =
n1 + 2
n2 + 2
∗ p̃i =
number of successes in the i th sample + 1
, i = 1, 2
ni + 2
• Inference about two proportions — p1 − p2
• Plus four confidence intervals:
∗
∗
∗ (p̃1 − p̃2 ) − z SE, (p̃1 + p̃2 ) + z SE , where SE is the
standard error of p̃1 − p̃2 :
s
p̃1 (1 − p̃1 ) p̃2 (1 − p̃2 )
+
SE =
n1 + 2
n2 + 2
number of successes in the i th sample + 1
, i = 1, 2
ni + 2
∗ Use it when n1 ≥ 5 and n2 ≥ 5
∗ p̃i =
• Test of significance:
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
1
n2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
∗ P-value:
1
n2
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
1
n2
∗ P-value:
? Ha : p1 − p2 > 0 — upper tail probability
corresponding to z
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
∗ P-value:
? Ha : p1 − p2 > 0
corresponding to
? Ha : p1 − p2 < 0
corresponding to
1
n2
— upper tail probability
z
— lower tail probability
z
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
∗ P-value:
? Ha : p1 − p2 > 0
corresponding to
? Ha : p1 − p2 < 0
corresponding to
? Ha : p1 − p2 6= 0
corresponding to
1
n2
— upper tail probability
z
— lower tail probability
z
— twice upper tail probability
|z|
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
1
n2
∗ P-value:
? Ha : p1 − p2 > 0 — upper tail probability
corresponding to z
? Ha : p1 − p2 < 0 — lower tail probability
corresponding to z
? Ha : p1 − p2 6= 0 — twice upper tail probability
corresponding to |z|
∗ significance level α and conclusion
• Test of significance:
∗ hypotheses: H0 v.s Ha / H0 : p1 = p2 (p1 − p2 = 0)
∗ pooled sample proportion p̂:
number of successes in both samples combined
p̂ =
number of individuals in both samples combined
p̂1 − p̂2
p̂(1 − p̂) n11 +
∗ test statistics: z = s
1
n2
∗ P-value:
? Ha : p1 − p2 > 0 — upper tail probability
corresponding to z
? Ha : p1 − p2 < 0 — lower tail probability
corresponding to z
? Ha : p1 − p2 6= 0 — twice upper tail probability
corresponding to |z|
∗ significance level α and conclusion
∗ use this test when counts of successes and failures are
each 5 or more in boh samples
• Chi-square test for a two-way table
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Compares the observed counts in the cells of the two-way
table with the counts that would be expected if H0 were true
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Compares the observed counts in the cells of the two-way
table with the counts that would be expected if H0 were true
expected count =
row total × column total
table total
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Compares the observed counts in the cells of the two-way
table with the counts that would be expected if H0 were true
expected count =
row total × column total
table total
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Compares the observed counts in the cells of the two-way
table with the counts that would be expected if H0 were true
expected count =
row total × column total
table total
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Degrees of freedom of χ2 : (r − 1)(c − 1), where r is the
number of rows and c is the number of columns
• Chi-square test for a two-way table
• Hypotheses: H0 : there is no relationship between the two
variables (row variable and column variable) v.s. Ha : there is
some relationship
• Compares the observed counts in the cells of the two-way
table with the counts that would be expected if H0 were true
expected count =
row total × column total
table total
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Degrees of freedom of χ2 : (r − 1)(c − 1), where r is the
number of rows and c is the number of columns
• P-value: the area under the chi-square density curve to the
right of the value of the test statistic
• Chi-square test for goodness of fit
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Compares the observed counts of each category with the
counts that would be expected if H0 were true
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Compares the observed counts of each category with the
counts that would be expected if H0 were true
expected count for category i = npi0
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Compares the observed counts of each category with the
counts that would be expected if H0 were true
expected count for category i = npi0
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Compares the observed counts of each category with the
counts that would be expected if H0 were true
expected count for category i = npi0
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Degrees of freedom of χ2 : k − 1, where k is the number of
categories
• Chi-square test for goodness of fit
• Null hypothesis: H0 : p1 = p10 , p2 = p20 , . . . , pk = pk0
• Compares the observed counts of each category with the
counts that would be expected if H0 were true
expected count for category i = npi0
• Chi-square test statistic:
χ2 =
X (observed count − expected count)2
expected count
• Degrees of freedom of χ2 : k − 1, where k is the number of
categories
• P-value: the area under the chi-square density curve to the
right of the value of the test statistic
• One-way analysis of variance (ANOVA) compares the means
of sevral populations.
• One-way analysis of variance (ANOVA) compares the means
of sevral populations.
• Hypotheses for ANOVA F -test: H0 : all the populations have
the same mean v.s. Ha : not all the means are the same
• One-way analysis of variance (ANOVA) compares the means
of sevral populations.
• Hypotheses for ANOVA F -test: H0 : all the populations have
the same mean v.s. Ha : not all the means are the same
•
F =
variation among the sample means
variation among individuals among the same sample
• One-way analysis of variance (ANOVA) compares the means
of sevral populations.
• Hypotheses for ANOVA F -test: H0 : all the populations have
the same mean v.s. Ha : not all the means are the same
•
F =
variation among the sample means
variation among individuals among the same sample
degrees of freedom for the numerator is I − 1 and degrees of
freedom for the denominator is N − I , where I is the number
of populations and N is the total number of observations from
I samples
• One-way analysis of variance (ANOVA) compares the means
of sevral populations.
• Hypotheses for ANOVA F -test: H0 : all the populations have
the same mean v.s. Ha : not all the means are the same
•
F =
variation among the sample means
variation among individuals among the same sample
degrees of freedom for the numerator is I − 1 and degrees of
freedom for the denominator is N − I , where I is the number
of populations and N is the total number of observations from
I samples
• Conditions for use ANOVA: independent SRS from each
population; each population is Normally distributed; all
populations have the same standard deviation