Review
Review
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
Review
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
Review
• Inference about µ with known σ — z-procedures (confidence
interval & test of significance)
• Confidence intervals:
∗ form: estimate ± margin of error / interpretation
Review
• 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
Review
• 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
• 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)