Summary of Formulas/Tests (Chapter 8 & Chapter 9)
Are you working hard on formulas?
 Note: All the H0 and H1 in this summary are just examples.)
Chapter 8: Hypothesis Testing
1. Testing a claim about a proportion:
H0: P = 0.5
H1: p > 0.5
Test statistic: 𝑧 =
𝑝̂−𝑝
𝑝𝑞
𝑛
√
Critical value: the standard normal distribution (Table A-2)
Requirements:
 Simple random samples
 Binomial distribution
 np ≥ 5 and nq ≥ 5
Note: P is the assumed proportion used in the claim, not the sample proportion.
2. Testing a claim about a population mean (with σ known):
H0: µ = 58
H1: µ ≠ 58
Test statistic: 𝑧 =
𝑥−µ
𝜎
√𝑛
Critical value: the Standard normal distribution (Table 2)
Requirements:
 σ in known.
 Simple random samples
 Normally distributed or n > 30
3. Testing a claim about a mean (with σ not known but s known):
H0: µ = 0.83
H1: µ < 0.83
Test statistic: 𝑡 =
𝑥−µ
𝑠
√𝑛
Critical value: t distribution (Table A-3 with df = n – 1)
Requirements:
 σ is not known but s is known
 Simple random samples
 Normally distributed or n > 30
4. Testing a claim about a standard deviation or variance:
H0: σ = 0.051
H1: σ < 0.051
Test statistic: 𝑥² =
(𝑛−1)𝑠²
𝜎²
Critical value: Chi-square (x²) distribution (Table A-4 with df = n – 1)
Requirements:
 Simple random sample
 Population normally distributed.
Chapter 9: Inferences from Two Samples
1. Inferences about two proportions:
H0: P1 = P2
H1: P1 > P2
𝑝̂ 1=
𝑥1
𝑛1
and 𝑝̂ 2=
𝑥2
𝑛2
Pooled sample proportion: 𝑃 =
𝑋1+𝑋2
𝑛1+𝑛2
The complement of 𝑃: 𝑞 = 1 − 𝑝
Test statistic: 𝑧 =
(𝑝̂1−𝑞̂1)−(𝑝1−𝑝2)
√𝑝 𝑞+𝑝 𝑞
𝑛1
where p1 – p2 =0 (assumed in the null hypothesis)
𝑛2
Critical value: the standard normal distribution (Table A-2)

Confidence interval estimate of p1 – p2:
Margin of error: 𝐸 = 𝑍𝛼/2√
𝑝̂1𝑞̂1
𝑛1
+
𝑝̂2𝑞̂2
𝑛2
(zα/2: Table A-2)
Confidence interval: (𝑝̂ 1 − 𝑝̂ 2) − 𝐸 < (𝑝1 − 𝑝2) < (𝑝̂ 1 − 𝑝̂ 2) + 𝐸
Requirements:
 We have proportions from two simple random samples that are independent.
 For each of the two samples, the number of successes is at least 5 and the
number of failures is at least 5.
2. Inference about two means: independent samples with σ1 and σ2 unknown and not
assumed equal
H0: µ1 = µ2
H1: µ1 > µ2
Test statistic: 𝑡 =
(𝑥1−𝑥2)−(µ1−µ2)
2
2
𝑛1
𝑛2
√𝑆1 +𝑆2
Critical value: t distribution (Table A-3 with df = smaller of n1 – 1 and n2 – 1)

Confidence interval estimate of µ1 - µ2:
𝑆1²
Margin of error: 𝐸 = 𝑡𝛼/2√ 𝑛1 +
𝑆2²
𝑛2
(tα/2: df = smaller of n1 – 1 and n2 – 1)
Confidence interval: ( 𝑥1 − 𝑥2) − 𝐸 < (µ1 − µ2) < ( 𝑥1 − 𝑥2) + 𝐸
Requirements:
 σ1nd σ2 are unknown and not assumed equal
 The two samples are independent and simple random samples.
 n1 > 30 and n2 > 30 or both samples come from populations having
normal distributions.
3. Inference about two means: independent samples with σ1 and σ2 known:
H0: µ1 = µ2
H1: µ1 > µ2
Test statistic: 𝑧 =
(𝑥1−𝑥2)−(µ1−µ2)
2
2
𝑛1
𝑛2
√𝜎1 +𝜎2
Critical value: the standard normal distribution (Table A-2)

Confidence interval estimate of µ1 - µ2:
𝜎1²
Margin of error: 𝐸 = 𝑍𝛼/2√
𝑛1
+
𝜎2²
𝑛2
(zα/2: Table A-2)
Confidence interval: ( 𝑥1 − 𝑥2) − 𝐸 < (µ1 − µ2) < ( 𝑥1 − 𝑥2) + 𝐸
Requirements:
 σ1 and σ2 both known.
 The two samples are independent and simple random samples.
 n1 > 30 and n2 > 30 or both samples come from populations having normal
distributions.
4. Inference about two means: independent samples with σ1 and σ2 unknown but they are
assumed to be equal
H0: µ1 = µ2
H1: µ1 > µ2
Pooled variance: 𝑆𝑝² =
Test statistic: 𝑡 =
(𝑛1−1)𝑆1²+(𝑛2−1)𝑆2²
(𝑛1−1)+(𝑛2−1)
(𝑥1−𝑥2)−(µ1−µ2)
√𝑆𝑝²+𝑆𝑝²
𝑛1
𝑛2
Critical value: Critical value: t distribution (Table A-3 with df = n1 + n2 -2)

Confidence interval estimate of µ1 - µ2:
𝑆𝑝²
Margin of error: 𝐸 = 𝑡𝛼/2√ 𝑛1 +
𝑆𝑝²
𝑛2
(tα/2: Table A-3 with df = n1 + n2 -2)
Confidence interval: ( 𝑥1 − 𝑥2) − 𝐸 < (µ1 − µ2) < ( 𝑥1 − 𝑥2) + 𝐸
Requirements:
 σ1 and σ2 are unknown but they are assumed to be equal.
 The two samples are independent and simple random samples.
 n1 > 30 and n2 > 30 or both samples come from populations having normal
distributions.
5. Inference about matched pairs:
H0: µd = 0
H1: µd ≠ 0
Test statistic: 𝑡 =
𝑑−µ𝑑
𝑆𝑑
√𝑛
Critical value: t distribution (Table A-3 with df = n – 1)

Confidence interval for matched pairs:
Margin of error: 𝐸 = 𝑡𝛼/2
𝑆𝑑
√𝑛
(tα/2: Table A-3 with df = n – 1)
Confidence interval: 𝑑 − 𝐸 < µ𝑑 < 𝑑 + 𝐸

Notations:
d: individual difference between the two values in a single matched pair
µd: mean value of the differences d for the population of all matched pairs
𝑑: mean value of the differences d for the paired sample data
Sd: standard deviation of the differences d for the paired sample data
n: number of pairs of data
Requirements:
 Matched pairs.
 The samples are simple random samples.
 The number of matched pairs of sample data is large (n>30) or the pairs of
values have differences that are from a population having a distribution that is
approximately normal.
6. Comparing variation in two samples:
H0: σ1² = σ2²
H1: σ1² ≠ σ2²
𝑆1²
Test statistic: 𝐹 = 𝑆2² (where s1² is the larger of the two sample variances)
Critical value: F distribution (Table A-5 with numerator df = n1 – 1 and denominator
df = n2 – 1)
Requirements:
 Two independent samples
 The two populations are each normally distributed.