statistic 1

Sheet1

Country	Life Expectancy
Angola	63.24
Australia	83.73
Bangladesh	73.98
Belgium	82.46
Bolivia	68.78
Burkina Faso	60.57
Cameroon	61.92
Canada	82.58
Chile	81.16
China	78.79
Congo	63.26
Croatia	79.4
Denmark	82.58
Djibouti	63.71
Domincan Republic	74.36
Egypt	70.81
Ethiopia	66.65
Fiji	68.45
Grenada	75.49
Iceland	83.02
Iran	76.97
Italy	84.2
Japan	84.59
Jordan	75.02
Latvia	76.06
Madagascar	66.43
Malaysia	76.42
Mali	60.03
Mongolia	72.94
Nepal	70.78
Netherlands	82.58
New Zealand	83.16
Nigeria	53.37
Oman	78.97
Pakistan	67.34
Paraguay	74.1
Peru	76.96
Russia	74.57
Samoa	72.75
Senegal	69.31
Singapore	84.27
South Korea	84.14
Sri Lanka	76.8
Sweden	83.65
Turkey	78.68
Uganda	63.84
United Kingdon	82.31
United States	79.74
Vietnam	74.74
Zimbabwe	61.92

Name Symbol Formula Description

Basic Statistics

Sample Size 𝑛 n/a

The number of data points in

a sample.

Sample Proportion �̂� �̂� = 𝑥

𝑛

The proportion (percentage)

from a sample.

Population Proportion 𝑝 𝑝 = 𝑥

𝑁

The total proportion

(percentage) from a

population.

Sample Mean �̅� �̅� = ∑ 𝑥

𝑛

The arithmetic average of a

sample.

Population Mean 𝜇 𝜇 = ∑ 𝑥

𝑁

The arithmetic average of a

population.

Sample Standard

Deviation 𝑠 𝑠 = √∑

(𝑥 − �̅�)2

𝑛

The standard deviation from

a sample.

Population Standard

Deviation 𝜎 𝜎 = √∑

(𝑥 − 𝜇)2

𝑁

The standard deviation from

a population.

Sample Variance 𝑠2 𝑠2 = ∑ (𝑥 − �̅�)2

𝑛 The variance from a sample.

Population Variance 𝜎2 𝜎2 = ∑ (𝑥 − 𝜇)2

𝑁

The variance from a

population.

Critical Values

Critical Values

𝑧𝛼 2⁄

𝑧-score found at the 𝛼 2⁄ th percentile of a

standard normal distribution

Critical value used for

confidence intervals

estimating a population

proportion or for two-tailed

hypothesis tests about a

population proportion.

Critical Values

𝑧𝑎

𝑧-score found at the 𝛼th percentile of a

standard normal distribution

Critical value used for left-

tailed or right-tailed

hypothesis tests about a

population proportion.

𝑡𝛼 2⁄

𝑡-statistic found at the 𝛼 2⁄ th percentile of

a student’s 𝑡 distribution (𝑛 degrees of

freedom)

Critical value used for

confidence intervals

estimating a population mean

or for two-tailed hypothesis

tests about a population mean.

𝑡𝛼

𝑡-statistic found at the 𝛼th percentile of a

student’s 𝑡 distribution (𝑛 degrees of

freedom)

Critical value used for left-

tailed or right-tailed

hypothesis tests about a

population mean.

𝜒𝐿 2

Chi-squared statistic found at the bottom

𝛼th percentile of a Chi-squared

distribution (𝑛 degrees of freedom)

Left critical value used in a

confidence intervals and

hypothesis tests about a

population standard deviation.

𝜒𝑅 2

Chi-squared statistic found at the top 𝛼th

percentile of a Chi-squared distribution (𝑛

degrees of freedom)

Right critical value used in a

confidence intervals and

hypothesis tests about a

population standard deviation.

𝑟

𝑟 = 𝑡𝛼 2⁄

√(𝑡𝛼 2⁄ ) 2

+ (𝑛 − 2)

(𝑛 − 2 degrees of freedom)

Critical Values of the Pearson

Correlation Coefficient 𝑟.

Margins of Error

Margin of Error 𝐸

𝐸 = 𝑧𝛼 2⁄ √ �̂� �̂�

𝑛

Margin of error for a

confidence interval estimating

a population proportion.

𝐸 = 𝑡𝛼 2⁄

𝑠

√𝑛

Margin of error for a

confidence interval estimating

a population mean.

Confidence Intervals

Confidence Interval

Confidence Interval

n/a

�̂� − 𝐸 < 𝑝 < �̂� + 𝐸

Confidence interval for

estimating a population

proportion.

�̅� − 𝐸 < 𝜇 < �̅� + 𝐸 Confidence interval for

estimating a population mean.

√ (𝑛 − 1)𝑠2

𝜒𝑅 2 < 𝜎 < √

(𝑛 − 1)𝑠2

𝜒𝐿 2

Confidence interval for

estimating a population

standard deviation.

Hypothesis Tests

Hypothesis Test on One

Sample - Proportions n/a

𝐻0: 𝑝 = 𝑋

𝐻𝑎: 𝑝 < 𝑋 Claim uses “less than”

𝐻0: 𝑝 = 𝑋

𝐻𝑎: 𝑝 > 𝑋 Claim uses “greater than”.

𝐻0: 𝑝 = 𝑋

𝐻𝑎: 𝑝 ≠ 𝑋

Claim uses “equal to” or “not

equal to”.

Hypothesis Test on One

Sample – Means

n/a

n/a

𝐻0: 𝜇 = 𝑋

𝐻𝑎: 𝜇 < 𝑋

Claim uses “less than”

𝐻0: 𝜇 = 𝑋 𝐻𝑎: 𝜇 > 𝑋

Claim uses “greater than”.

𝐻0: 𝜇 = 𝑋

𝐻𝑎: 𝜇 ≠ 𝑋

Claim uses “equal to” or “not

equal to”.

Hypothesis Test on One

Sample – Standard

Deviation

n/a

𝐻0: 𝜎 = 𝑋

𝐻𝑎: 𝜎 < 𝑋 Claim uses “less than”

𝐻0: 𝜎 = 𝑋 𝐻𝑎: 𝜎 > 𝑋

Claim uses “greater than”.

𝐻0: 𝜎 = 𝑋

𝐻𝑎: 𝜎 ≠ 𝑋

Claim uses “equal to” or “not

equal to”.

Hypothesis Test on Two

Samples – Proportions

Hypothesis Test on Two

Samples - Proportions

n/a

𝐻0: 𝑝1 = 𝑝2

𝐻𝑎: 𝑝1 < 𝑝2 Claim uses “less than”

𝐻0: 𝑝1 = 𝑝2

𝐻𝑎: 𝑝1 > 𝑝2 Claim uses “greater than”.

𝐻0: 𝑝1 = 𝑝2

𝐻𝑎: 𝑝1 ≠ 𝑝2

Claim uses “equal to” or

“not equal to”.

Hypothesis Test on Two

Samples – Independent

Means

n/a

𝐻0: 𝜇1 = 𝜇2

𝐻𝑎: 𝜇1 < 𝜇2 Claim uses “less than”

𝐻0: 𝜇1 = 𝜇2

𝐻𝑎: 𝜇1 > 𝜇2 Claim uses “greater than”.

𝐻0: 𝜇1 = 𝜇2

𝐻𝑎: 𝜇1 ≠ 𝜇2

Claim uses “equal to” or “not

equal to”.

Hypothesis Test on Two

Samples – Dependent

Means

n/a

n/a

𝐻0: 𝜇𝑑 = 𝑋

𝐻𝑎: 𝜇𝑑 < 𝑋 Claim uses “less than”

𝐻0: 𝜇𝑑 = 𝑋

𝐻𝑎: 𝜇𝑑 > 𝑋 Claim uses “greater than”.

𝐻0: 𝜇𝑑 = 𝑋

𝐻1: 𝜇𝑑 ≠ 𝑋

Claim uses “equal to” or “not

equal to”.

Test Statistics

𝑧 𝑧 =

�̂� − 𝑝

√ 𝑝𝑞 𝑛

Test statistic used for

hypothesis test about a

population proportion.

Test Statistic

Test Statistic

𝑧 = (�̂�1 − �̂�2) − (𝑝1 − 𝑝2)

√ �̅� �̅� 𝑛1

+ �̅� �̅� 𝑛2

�̅� = 𝑥1 + 𝑥2

𝑛1 + 𝑛2 , �̅� = 1 − �̅�

Test statistic used for a

hypothesis test about two

population proportions.

Pooled proportion.

𝑡

𝑡 = �̅� − 𝜇

𝑠

√𝑛

Test statistic used for a

hypothesis test about a

population mean.

𝑡 = (�̅�1 − �̅�2) − (𝜇1 − 𝜇2)

√ 𝑠1

2

𝑛1 +

𝑠2 2

𝑛2

Test statistic used for a

hypothesis test about two

independent population

means.

𝑡 = �̅� − 𝜇𝑑

𝑠𝑑

√𝑛

Test statistic used for a

hypothesis test about two

dependent means.

𝜒2

𝜒2 = (𝑛 − 1)𝑠2

𝜎2

Test statistic used for a

hypothesis test about a

population standard deviation.

𝜒2 = ∑ (𝑂 − 𝐸)2

𝐸

Test statistic used for a

goodness of fit test or a test of

independence.

Independent Samples

Sample Average

Difference �̅� �̅� = ∑

𝑑

𝑛

The arithmetic average of a

set of differences of matched

pairs from a sample.

Population Average

Difference 𝜇𝑑 𝜇𝑑 = ∑

𝑑

𝑁

The arithmetic average of a

set of differences of matched

pairs from a population.

Sample Standard

Deviation of Differences 𝑠𝑑 𝑠𝑑 = √∑

(𝑑 − �̅�) 2

𝑛

The standard deviation of a

set of differences of matched

pairs.

Linear Regression

Regression Equation

n/a �̂� = 𝑏0 + 𝑏1𝑥 Line of best fit.

�̂�

Substitute dependent variable into

regression equation.

Best predicted value when

regression equation is a good

model.

�̅� �̅� = ∑ 𝑥

𝑛

Average of 𝑦 values, best

predicted value when

regression equation is not a

good model.

𝑏0 𝑏1 = 𝑛(∑ 𝑥𝑦) − (∑ 𝑥)(∑ 𝑦)

𝑛(∑ 𝑥2) − (∑ 𝑥)2 𝑦 − intercept

𝑏1 𝑏0 = (∑ 𝑦)(∑ 𝑥2) − (∑ 𝑥)(∑ 𝑥𝑦)

𝑛(∑ 𝑥2) − (∑ 𝑥)2 slope

Correlation Coefficient 𝑟

𝑟

= 𝑛(∑ 𝑥𝑦) − (∑ 𝑥)(∑ 𝑦)

√𝑛(∑ 𝑥2) − (∑ 𝑥)2√𝑛(∑ 𝑦2) − (∑ 𝑦)2

Measures the strength of the

linear correlation between two

sets of paired variables.

Coefficient of

Determination 𝑟2 Square of the Correlation Coefficient, 𝑟

Measures the ratio of

explained variance to total

variance between two sets of

paired variables.