Statistics_Hypothesis_Testing(p-value)

Hypothesis Testing MAT-150: Statistics 1 Spring 2015 Page: 1(P-Value Method)

Definition

Textbook Definition: The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.

The difference between the traditional method and the p-value method:

The traditional method compares the test value/statistic to the critical value The p-value method compares the p-value (a probability) to the significance level (α value)

Determine which Test

z-Test for Mean

When sigma (σ), a.k.a the population standard deviation, is known (given) and the problem asks to test a claim dealing with a mean

z-Test for Proportion

When the problem asks to test a claim dealing with a proportion; if the problem is talking about percentages, or mentions some type of proportion (i.e. 84 students out of a random sample of 100 BCC students passed their math course), this test will most likely be used

t-Test for Mean

When sigma (σ), a.k.a the population standard deviation, is unknown (not given), but instead the sample standard deviation (s) is known (given) and the problem asks to test a claim dealing with a mean

Χ2-Test for Variance or Standard Deviation

When the problem asks to test a claim dealing with a population variance (σ2) or a population standard deviation (σ) [recall: variance = (standard deviation)2]

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Gather information from problem & check assumptions

z-Test for Mean

µ = population mean

x = sample mean ¿

σ = population standard deviation

n = sample size

α = level of significance

Assumptions1. The sample is a random sample2. Either n ≥ 30 or the population is normally distributed if n ¿ 30

z-Test for Proportion

p = population proportion

q = 1 – p

p̂ = sample proportion ¿

n = sample size

α = level of significance

Assumptions1. The sample is a random sample2. The conditions for a binomial experiment are satisfied [page 437 8th edition: explanation; refer to chapter 5 section 3 page 271)3. np ≥ 5 and nq ≥ 5

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t-Test for Mean

µ = population mean

x = sample mean ¿

s = sample standard deviation ¿

n = sample size

n-1 = degrees of freedom

α = level of significance

Assumptions1. The sample is a random sample2. Either n ≥ 30 or the population is normally distributed if n ¿ 30

Χ2-Test for Variance or Standard Deviation

σ2 = population variance [remember: population variance = (population standard deviation)2

s2 = sample variance [remember: sample variance = (sample standard deviation)2 ¿

n = sample size

n-1 = degrees of freedom

α = level of significance

Assumptions1. The sample must be randomly selected from the population2. The population must be normally distributed for the variable under study 3. The observations must be independent of one another

Procedure for each type of Test (P-value method)

Steps for z-Test (Mean)

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1. State the hypotheses [ null hypothesis (Ho) & alternative hypothesis (H1 or Ha) ] and identify the claim:

Ho: µ ("=") some population mean i. Note: depending on professor, the symbol used for the null hypothesis will

always be "=" OR the symbol could either be one of the following options "=" ; "≥" ; "≤"

1. If professor prefers using one of the three symbols ("=" ; "≥" ; "≤"), in order to determine which one to use, refer to the alternative hypothesis and use the opposite symbol

a. The opposite symbols for "≠" is "="i. Example: Ho: µ = 100

Ha: µ ≠ 100b. The opposite symbols for "¿" is "≤"

i. Example: Ho: µ ≤ 100 Ha: µ ¿ 100

c. The opposite symbols for "¿" is "≥"i. Example: Ho: µ ≥ 100

Ha: µ ¿ 100ii. Note: the null hypothesis will always contain some type of equality symbol

1. Refer to the table below for the different common phrases for "=" ; "≥" ; "≤"

Ha: µ ("≠" ; "¿" ; "¿") some population meani. Note: in order to determine which symbol to use look out for common phrases

(refer to table below)ii. Note: if "¿" (right-tailed) or "¿" (left-tailed) is used, the test is a one-tailed test;

if "≠" is used, the test becomes a two-tailed test

Common Phrases used for Hypothesis Testing> < = ≠

is greater than is less than is equal to is not equal tois above is below is the same as is different from

is higher than is lower than has not changed from has changed fromis longer than is shorter than is the same is not the sameis increased is smaller than is “is not”is too low Is decreased or reduced is equivalent to

Is too high

≥ ≤no less than no more than

at least at most

2. Compute the test value/statistic (a.k.a the z-score): Write down all information given in problem (refer to "gather information section")

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Plug information into formula: z = x−μσ

√n OR

(x−μ)√nσ

3. Find the p-value: Easiest way to find the p-value for z-score (your test statistic), which was found in the

previous step:i. Refer to the z-table (Table E)

ii. Look up the negative value (if not already negative) of the z-score (test statistic) on the z-table (Table E) in order to find the corresponding proportion

iii. This proportion (that was found inside Table E) is your p-value1. Note: if the test is a two-tailed test, then you have to double the p-value

4. Make the decision to either "reject" or "do not reject" the null hypothesis (Ho): If p-value ≤ α : reject Ho

If p-value ¿ α : do not reject Ho

5. Summarize Results: There are four different possibilities for the conclusion:

i. If the claim is in Ho and Ho is rejected: There is enough/sufficient evidence to reject the claim that …

ii. If the claim is in Ho and Ho is not rejected: There is not enough/insufficient evidence to reject the claim that …

iii. If the claim is in H1 and Ho is rejected: There is enough/sufficient evidence to support the claim that …

iv. If the claim is in H1 and Ho is not rejected: There is not enough/insufficient evidence to support the claim that …

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Steps for z-Test (Proportion)

1. State the hypotheses [ null hypothesis (Ho) & alternative hypothesis (H1 or Ha) ] and identify the claim:

a. Ho: p ("=") some population mean i. Note: depending on professor, the symbol used for the null hypothesis will

always be "=" OR the symbol could either be one of the following options "=" ; "≥" ; "≤"

1. If professor prefers using one of the three symbols ("=" ; "≥" ; "≤"), in order to determine which one to use, refer to the alternative hypothesis and use the opposite symbol

a. The opposite symbols for "≠" is "="i. Example: Ho: p = 100

Ha: p ≠ 100b. The opposite symbols for "¿" is "≤"

i. Example: Ho: p ≤ 100 Ha: p ¿ 100

c. The opposite symbols for "¿" is "≥"i. Example: Ho: p ≥ 100

Ha: p ¿ 100ii. Note: the null hypothesis will always contain some type of equality symbol

1. Refer to the table below for the different common phrases for "=" ; "≥" ; "≤"

b. Ha: p ("≠" ; "¿" ; "¿") some population proportion i. Note: in order to determine which symbol to use look out for common phrases

(refer to table below)ii. Note: if "¿" (right-tailed) or "¿" (left-tailed) is used, the test is a one-tailed test;

if "≠" is used, the test becomes a two-tailed test

Common Phrases used for Hypothesis Testing> < = ≠

is greater than is less than is equal to is not equal tois above is below is the same as is different from

is higher than is lower than has not changed from has changed fromis longer than is shorter than is the same is not the sameis increased is smaller than is “is not”is too low Is decreased or reduced is equivalent to

Is too high

≥ ≤no less than no more than

at least at most

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2. Compute the test value/statistic (a.k.a the z-score):a. Write down all information given in problem (refer to "gather information section")

b. Plug information into formula: z = p̂−p

√ p ∙qn OR ( p̂−p ) √n

√ p ∙q3. Find the p-value:

a. Easiest way to find the p-value for z-score (your test statistic), which was found in the previous step:

i. Refer to the z-table (Table E)ii. Look up the negative value (if not already negative) of the z-score (test

statistic) on the z-table (Table E) in order to find the corresponding proportioniii. This proportion (that was found inside Table E) is your p-value

1. Note: if the test is a two-tailed test, then you have to double the p-value

4. Make the decision to either "reject" or "do not reject" the null hypothesis (Ho):a. If p-value ≤ α : reject Ho

b. If p-value ¿ α : do not reject Ho

5. Summarize Results:a. There are four different possibilities for the conclusion:

i. If the claim is in Ho and Ho is rejected: There is enough/sufficient evidence to reject the claim that …

ii. If the claim is in Ho and Ho is not rejected: There is not enough/insufficient evidence to reject the claim that …

iii. If the claim is in H1 and Ho is rejected: There is enough/sufficient evidence to support the claim that …

iv. If the claim is in H1 and Ho is not rejected: There is not enough/insufficient evidence to support the claim that …

***NOTICE: the differences between z-Test (Mean) and z-Test (Proportion): "p" is used in the

hypotheses and the formula used is z = p̂−p

√ p ∙qn (so, make sure you write down the correct

information)***

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t-Test for Mean

1. State the hypotheses [ null hypothesis (Ho) & alternative hypothesis (H1 or Ha) ] and identify the claim:

Ho: µ ("=" OR "=" ; "≥" ; "≤") some population mean i. Note: depending on professor, the symbol used for the null hypothesis will

always be "=" OR the symbol could either be one of the following options "=" ; "≥" ; "≤"

1. If professor prefers using one of the three symbols ("=" ; "≥" ; "≤"), in order to determine which one to use, refer to the alternative hypothesis and use the opposite symbol

a. The opposite symbols for "≠" is "="i. Example: Ho: µ = 100

Ha: µ ≠ 100b. The opposite symbols for "¿" is "≤"

i. Example: Ho: µ ≤ 100 Ha: µ ¿ 100

c. The opposite symbols for "¿" is "≥"i. Example: Ho: µ ≥ 100

Ha: µ ¿ 100ii. Note: the null hypothesis will always contain some type of equality symbol

1. Refer to the table below for the different common phrases for "=" ; "≥" ; "≤"

Ha: µ ("≠" ; "¿" ; "¿") some population meani. Note: in order to determine which symbol to use look out for common phrases

(refer to table below)ii. Note: if "¿" (right-tailed) or "¿" (left-tailed) is used, the test is a one-tailed test;

if "≠" is used, the test becomes a two-tailed test

Common Phrases used for Hypothesis Testing> < = ≠

is greater than is less than is equal to is not equal tois above is below is the same as is different from

is higher than is lower than has not changed from has changed fromis longer than is shorter than is the same is not the sameis increased is smaller than is “is not”is too low Is decreased or reduced is equivalent to

Is too high

≥ ≤no less than no more than

at least at most

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2. Compute the test value/statistic (a.k.a the t-score): Write down all information given in problem (refer to "gather information section")

Plug information into formula: t = x−μs

√n OR

(x−μ)√ns

3. Find the p-value: t-Tests involve p-value intervals; in order to find this interval for a t-score (your test

statistic), which was found in the previous step:i. Refer to the t-table (Table F)

ii. Find the row that corresponds to the problem’s degrees of freedom (d.f = n-1)iii. Find the two values that your t-score (test statistic) falls between iv. Look up to the row labeled one-tail or two-tail to find α values for your p-value

interval (refer to row that corresponds to the problem)1. Recall: If "¿" or "¿" is used, the test is a one-tailed test

If "≠" is used, the test becomes a two-tailed testv. Create your p-value interval with the two α values found:

( lower bound /number )<¿ p-value ¿ (upper bound /number ) vi. For example (refer to picture below for visual help): Find the p-value when the

test statistic (t-score) is 2.056, the sample size is 11, and the test is right-tailed 1. Refer to Table F2. Next, go to row with d.f = 10 (n-1 = 11 - 1)3. Find two values that 2.056 falls between: it falls between 1.812 and

2.228 4. Then, look up to the row labeled one-tail since the problem is a right-

tailed testa. The two values are 0.05 and 0.025 which create your p-value

interval: 0.025 ¿ p-value ¿ 0.05

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4. Make the decision to either "reject" or "do not reject" the null hypothesis (Ho): If upper bound of p-value ≤ α : reject Ho

If lower bound of p-value ¿ α : do not reject Ho

i. For example: If p-value interval is 0.025 ¿ p-value ¿ 0.05 and α = .11. We would reject Ho because 0.05 (upper bound) ≤ .1

ii. For example: If p-value interval is 0.025 ¿ p-value ¿ 0.05 and α = 0.0051. We do not reject Ho because 0.025 (lower bound) ¿ 0.005

5. Summarize Results: There are four different possibilities for the conclusion:

i. If the claim is in Ho and Ho is rejected: There is enough/sufficient evidence to reject the claim that …

ii. If the claim is in Ho and Ho is not rejected: There is not enough/insufficient evidence to reject the claim that …

iii. If the claim is in H1 and Ho is rejected: There is enough/sufficient evidence to support the claim that …

iv. If the claim is in H1 and Ho is not rejected: There is not enough/insufficient evidence to support the claim that …

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Χ2-Test for Variance or Standard Deviation

1. State the hypotheses [ null hypothesis (Ho) & alternative hypothesis (H1 or Ha) ] and identify the claim:

Ho: σ 2 ("=" OR "=" ; "≥" ; "≤") some population variance OR Ho: σ ("=" OR "=" ; "≥" ; "≤") some population standard deviation

i. Note: depending on professor, the symbol used for the null hypothesis will always be "=" OR the symbol could either be one of the following options "=" ; "≥" ; "≤"

1. If professor prefers using one of the three symbols ("=" ; "≥" ; "≤"), in order to determine which one to use, refer to the alternative hypothesis and use the opposite symbol

a. The opposite symbols for "≠" is "="i. Example: Ho: σ 2 = 100 OR Ho: σ = 100

Ha: σ 2 ≠ 100 OR Ha: σ ≠ 100b. The opposite symbols for "¿" is "≤"

i. Example: Ho: σ 2 ≤ 100 OR Ho: σ ≤ 100

Ha: σ 2 ¿ 100 OR Ha: σ ¿ 100c. The opposite symbols for "¿" is "≥"

i. Example: Ho: σ 2 ≥ 100 OR Ho: σ ≥ 100

Ha: σ 2 ¿ 100 OR Ha: σ ¿ 100ii. Note: the null hypothesis will always contain some type of equality symbol

1. Refer to the table below for the different common phrases for "=" ; "≥" ; "≤"

Ha: σ 2 ("≠" ; "¿" ; "¿") some population variance

OR Ha: σ ("≠" ; "¿" ; "¿") some population standard

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i. Note: in order to determine which symbol to use look out for common phrases (refer to table below)

ii. Note: if "¿" (right-tailed) or "¿" (left-tailed) is used, the test is a one-tailed test; if "≠" is used, the test becomes a two-tailed test

Common Phrases used for Hypothesis Testing> < = ≠

is greater than is less than is equal to is not equal tois above is below is the same as is different from

is higher than is lower than has not changed from has changed fromis longer than is shorter than is the same is not the sameis increased is smaller than is “is not”is too low Is decreased or reduced is equivalent to

Is too high

≥ ≤no less than no more than

at least at most

2. Compute the test value/statistic (a.k.a the Χ2-score): Write down all information given in problem (refer to "gather information section")

Plug information into formula: Χ2 = (n−1 ) s2

σ 2

3. Find the p-value: Χ2-Tests involve p-value intervals; in order to find this interval for a Χ2-score (your test

statistic), which was found in previous step:i. Refer to the Χ2-table (Table G)

ii. Find the row that corresponds to the problem’s degrees of freedom (d.f = n-1)iii. Find the two values that your Χ2-score (test statistic) falls between iv. Look up to the top row to find α values for your p-value interval

1. If right-tailed test (α values on top row from 0.10 to 0.005), these α values will create p-value interval

a. Recall: "¿" means right-tailed test 2. If left-tailed test (α values on top row from 0.995 to 0.90), you need to

subtract each α value from 1; those values will create p-value intervala. Recall: "¿" means left-tailed test

3. If two-tailed test, each α value needs to be doubled; those values will create p-interval

a. Recall: "≠" means two-tailed testb. Note: If test statistic (Χ2-score) falls on the right side of the

distribution (top row values from 0.10 to 0.005), double each α value

c. Note: If test statistic (Χ2-score) falls on the left side of the distribution (top row values from 0.995 to 0.90), first subtract each α value from 1, then double it

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v. Create your p-value interval with the two α values found: ( lower bound /number )<¿ p-value ¿ (upper bound /number )

vi. Examples for each situation are on the next couple pages1. NOTICE: p-value intervals for Χ2-Tests will usually have values between

0.005 and 0.10 if it's a one-tailed test; if it's a two-tailed test, the values will be between 0.01 and 0.20

a. Note: This is valid for Table G that's used in Bluman's textbook2. TRICK: make the top row of Table G (Χ2 Table), starting from left side,

0.005, 0.01, 0.025, 0.05, 0.10, 0.10, 0.05, 0.025, 0.01, 0.005 (refer to picture below)

a. If you choose to use this trick, you will not have to worry about subtracting from "1"

i. However, if two-tailed you still need to double the values before creating your p-value interval

1. Example involving right tail (refer to picture below for visual help): Find the p-value when the test statistic (Χ2-score) is 19.274, the sample size is 8, and the test is right-tailed

a. First, refer to Table G b. Next, go to row with d.f = 7 (n-1 = 8 - 1)c. Find two values that 19.274 falls between: it falls between 18.475 and 20.278 d. Then, look up to the top row and since it’s a right-tailed test, these values will construct

your p-value interval i. The two values are 0.01 and 0.005 so the p-value interval is

0.005 ¿ p-value ¿ 0.01

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2. Example involving left tail (refer to picture below for visual help): Find the p-value when the test statistic (Χ2-score) is 2.940, the sample size is 11, and the test is left-tailed

a. First, refer to Table G b. Next, go to row with d.f = 10 (n-1 = 11 - 1)c. Find two values that 2.940 falls between: it falls between 2.558 and 3.247 d. Then, look up to the top row and since it’s a left-tailed test, these values will need to be

subtracted from 1 in order to construct your p-value interval i. The two values are 0.99 and 0.975, so subtract these values from 1:

1. 1 - 0.99 = 0.012. 1 - 0.975 = 0.025

ii. The two values used for the p-value interval are 0.01 and 0.0251. So, the p-value interval is 0.01 ¿ p-value ¿ 0.025

3. Example involving two-tails on left side of the distribution (refer to picture below for visual help): Find the p-value when the test statistic (Χ2-score) is 0.521, the sample size is 8, and the test is two-tailed

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a. First, refer to Table G b. Next, go to row with d.f = 5 (n-1 = 6 - 1)c. Find two values that 0.521 falls between: it falls between 0.412 and 0.554 d. Then, look up to the top row and since it’s a two-tailed test on the left side of the

distribution, these values will need to be subtracted from 1, then doubled in order to construct your p-value interval

i. The two values are 0.995 and 0.99, so first subtract these values from 1, then double the values:

1. 1 - 0.995 = 0.005 then 2 (0.005) = 0.012. 1 - 0.99 = 0.01 then 2 (0.01) = 0.02

ii. The two values used for the p-value interval are 0.01 and 0.021. So, the p-value interval is 0.01 ¿ p-value ¿ 0.02

4. Example involving two-tails on right side of the distribution (refer to picture below for visual help): Find the p-value when the test statistic (Χ2-score) is 8.420, the sample size is 5, and the test is two-tailed

a. First, refer to Table G b. Next, go to row with d.f = 4 (n-1 = 5 - 1)c. Find two values that 8.420 falls between: it falls between 7.779 and 9.488 d. Then, look up to the top row and since it’s a two-tailed test on the right side of the

distribution, these values will need to be doubled in order to construct your p-value interval

i. The two values are 0.10 and 0.05, so double these values:1. 2 (0.10) = 0.202. 2 (0.05) = 0.10

ii. The two values used for the p-value interval are 0.20 and 0.101. So, the p-value interval is 0.10 ¿ p-value ¿ 0.20

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4. Make the decision to either "reject" or "do not reject" the null hypothesis (Ho):a. If upper bound of p-value ≤ α : reject Ho

b. If lower bound of p-value ¿ α : do not reject Ho

i. For example: If p-value interval is .005 ¿ p-value ¿ .01 and α = .11. We would reject Ho because .01 (upper bound) ≤ .1

ii. For example: If p-value interval .005 ¿ p-value ¿ .01 and α = .0011. We do not reject Ho because .005 (lower bound) ¿ .001

5. Summarize Results:a. There are four different possibilities for the conclusion:

i. If the claim is in Ho and Ho is rejected: There is enough/sufficient evidence to reject the claim that …

ii. If the claim is in Ho and Ho is not rejected: There is not enough/insufficient evidence to reject the claim that …

iii. If the claim is in H1 and Ho is rejected: There is enough/sufficient evidence to support the claim that …

iv. If the claim is in H1 and Ho is not rejected: There is not enough/insufficient evidence to support the claim that …

***NOTICE: Test values/statistics can either be a z-score, t-score, or Χ2-score; it depends on the type of problem***

Bibliography

Bluman, Allan G. "Hypothesis Testing." Elementary Statistics: A Step by Step Approach, Seventh Edition.

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7th ed. New York: McGrawhill Higher Education, 2009. N. pag. Print.

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