Confidence Intervals Summary Sheet

Name Conditions Calculation Interpretation

1-proprtion z-interval

1) The sampling distribution is approx normal b/c: 10ˆ pn

10)ˆ1( pn2) The data comes from a random sample 3) The observations are independent b/c N ≥ 10n

ˆ ˆ(1 )ˆ *

p pp z

n

I am ___% confident that my interval of ____ to ____ contains the population proportion (context).

t-interval

1) The sampling distribution is (approx) normal b/c: population is (approx) normal OR n ≥ 30 so CLT applies OR When raw data, boxplot contains no outliers

or strong skewness2) The data comes from a random sample3) The observations are independent b/c N ≥ 10n

*s

x tn

df = n – 1

I am ___% confident that my interval of ____(units) to ____(units) contains the population mean (context).

2-proportion z-interval

For a Random Sample and/or Survey:1) The sampling distribution is approx normal b/c:

1ˆ 5np 1ˆ(1 ) 5n p 2ˆ 5np 2ˆ(1 ) 5n p

2) The two samples are random & independent3) The observations are independent b/c N ≥ 10n 1 1 2 2

1 21 2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ *

p p p pp p z

n n

I am ___% confident that my interval of ____ to ____ contains the population difference in the proportions of (context).For an Experiment:

1) The sampling distribution is approx normal b/c: 1ˆ 5np 1ˆ(1 ) 5n p 2ˆ 5np 2ˆ(1 ) 5n p

2) The data comes from a randomized experiment

Name Conditions Calculation Interpretation

2-sample t-interval

For a Random Sample and/or Survey:1) The sampling distribution is (approx) normal

b/c: Both populations are (approx) normal CLT applies to both sample sizes When raw data, neither boxplot contains

outliers or strong skewness2) The two samples are random & independent3) The observations are independent b/c N ≥ 10n

2 21 2

1 11 2

*s s

x x tn n

df = from calculator

I am ___% confident that my interval of ____(units) to ____(units) contains the population difference in the means (context).For an Experiment:

1) The sampling distribution is (approx) normal b/c: Both populations are (approx) normal CLT applies to both sample sizes When raw data, neither boxplot contains

outliers or strong skewness2) The data comes from a randomized experiment

paired t-interval

For a Random Sample and/or Survey:1) The sampling distribution is (approx) normal

b/c: Population of differences (approx) normal n ≥ 30 so CLT applies When raw data, boxplot of differences

contains no outliers or strong skewness2) The data comes from a random sample3) The observations are independent b/c N ≥ 10n

*d

sx t

n

df = n – 1

I am ___% confident that my interval of ____(units) to ____(units) contains the population mean difference of (context).For an Experiment:

1) The sampling distribution is (approx) normal b/c: Population of differences (approx) normal n ≥ 30 so CLT applies When raw data, boxplot of differences

contains no outliers or strong skewness2) The data comes from a randomized experiment

Name Conditions Calculation Interpretation

linear regressiont-interval

1) The data comes from a random sample or randomized experiment.

2) The population association is linear: b/c the scatterplot appears linear assume it if no data or scatterplot given

3) The errors are independent: b/c the residual plot is scattered assume it if no data or residual plot given

4) The variation in y is the same everywhere: b/c the residual plot shows consistent

spread assume it if no data or residual plot given

5) The errors about the population line follow normal models: b/c the normal probability plot appears

linear OR

b/c the histogram of the residuals appears bell-shaped

assume it if no data, normal probability plot, or histogram given

11 * bb t SE

df = n – 2

I am ___% confident that my interval of ____(units) to ____(units) contains the population slope of the line (context).

Significance Tests Summary

Name Hypotheses Conditions Calculations Interpretation

1-propotion z-test

H0: p = p0

Ha: p < p0 (or > or ≠)where p0 is the hypothesized p

1) The sampling distribution is approx normal b/c: 10np (use p0)

(1 ) 10n p (use p0)2) The data comes from a random sample 3) The observations are independent b/c N ≥ 10n

0

0 0

ˆ

(1 )

p pz

p p

n

NOTE: use p0 in SE

B/c p-value < α, there is enough evidence to support the claim that (alternative in context). ORB/c p-value > α, there is not enough evidence…

t-test

H0: µ = µ0

Ha: µ < µ0 (or > or ≠)where µ0 is the hypothesized µ

1) The sampling distribution is (approx) normal b/c: population is (approx) normal OR n ≥ 30 so CLT applies OR When raw data, boxplot contains no outliers

or strong skewness2) The data comes from a random sample3) The observations are independent b/c N ≥ 10n

0xt

s

n

df = n – 1

B/c p-value < α, there is enough evidence to support the claim that (alternative in context). ORB/c p-value > α, there is not enough evidence…

2-proportion z-test

H0: p1 = p2

Ha: p1 < p2 (or > or ≠)Define & use letters instead of using 1 & 2

For a Random Sample and/or Survey:1) The sampling distribution is approx normal b/c:

1ˆ 5np 1ˆ(1 ) 5n p 2ˆ 5np 2ˆ(1 ) 5n p

2) The two samples are random & independent3) The observations are independent b/c N ≥ 10n

1 2

1 2

ˆ ˆ

1 1ˆ ˆ1pooled pooled

p pz

p pn n

NOTE: for a test we assume p1 = p2

so use special case formula for SE

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…

For an Experiment:1) The sampling distribution is approx normal b/c:

1ˆ 5np 1ˆ(1 ) 5n p 2ˆ 5np 2ˆ(1 ) 5n p

2) The data comes from a randomized experiment

Name Hypotheses Conditions Calculations Interpretation

2-sample t-test

H0: µ1 = µ2

Ha: µ1 < µ2 (or > or ≠)Define & use letters instead of using 1 & 2

For a Random Sample and/or Survey:1) The sampling distribution is (approx) normal

b/c: Both populations are (approx) normal CLT applies to both sample sizes When raw data, neither boxplot contains

outliers or strong skewness2) The two samples are random & independent3) The observations are independent b/c N ≥ 10n

1 2

2 21 2

1 2

x xt

s s

n n

df = from calc. OR

df = smaller of n1 – 1 or n2 – 1

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…

For an Experiment:1) The sampling distribution is (approx) normal

b/c: Both populations are (approx) normal CLT applies to both sample sizes When raw data, neither boxplot contains

outliers or strong skewness2) The data comes from a randomized experiment

paired t-test

H0: µ = 0 (usually)Ha: µ < 0 (or > or ≠)where µ0 is the hypothesized mean difference (which is usually 0)

For a Random Sample and/or Survey:1) The sampling distribution is (approx) normal

b/c: Population of differences (approx) normal n ≥ 30 so CLT applies When raw data, boxplot of differences

contains no outliers or strong skewness2) The data comes from a random sample3) The observations are independent b/c N ≥ 10n

0d

d

xt

s

n

where μ0 is usually 0df = n – 1

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…

For an Experiment:1) The sampling distribution is (approx) normal

b/c: Population of differences (approx) normal n ≥ 30 so CLT applies When raw data, boxplot of differences

contains no outliers or strong skewness2) The data comes from a randomized experiment

Name Hypotheses Conditions Calculations Interpretation

χ2-GOF test

H0: The distribution is the same as the hypothesized distribution.

Ha: The distribution is not the same as the hypothesized distribution.

For a Random Sample and/or Survey:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a random sample 3) Observations are independent b/c N ≥ 10n 2

2 O E

E

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…

For an Experiment:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a randomized experiment

χ2-HOP test

H0: ___ and ___ are distributed the same as each other.

Ha: ___ and ___ are not distributed the same as each other.

For a Random Sample and/or Survey:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a random sample 3) Observations are independent b/c N ≥ 10n4) The samples are independent

2

2 O E

E

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is notenough evidence…

For an Experiment:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a randomized experiment

χ2-TOI test

H0: ___ and ___ are not associated.

Ha: ___ and ___ are associated.

For a Random Sample and/or Survey:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a random sample 3) Observations are independent b/c N ≥ 10n 2

2 O E

E

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…

For an Experiment:1) The sampling distribution is χ2 because:

All expected counts are ≥ 1 At least 80% of expected counts are ≥ 5

2) Data comes from a randomized experiment

Name Hypotheses Conditions Calculations Interpretation

linear regressiont-test

H0: β = 0 (usually)Ha: β < 0 (or > or ≠)where β 0 is the hypothesized slope (which is usually 0, but can be another value)

1) The data comes from a random sample or randomized experiment.

2) The population association is linear: b/c the scatterplot appears linear assume it if no data or scatterplot given

3) The errors are independent: b/c the residual plot is scattered assume it if no data or residual plot given

4) The variation in y is the same everywhere: b/c the residual plot shows consistent

spread assume it if no data or residual plot given

5) The errors about the population line follow normal models: b/c the normal probability plot appears

linear OR

b/c the histogram of the residuals appears bell-shaped

assume it if no data, normal probability plot, or histogram given

1

1 0

b

bt

SE

where β0 is usually 0df = n – 2

B/c p-value < α, there is enough evidence to support the claim that (alternative in context).

OR

B/c p-value > α, there is not enough evidence…