General Steps in Hypothesis Testing

Page 1

STEP 1:

• State the hypotheses

Null hypothesis • denoted by H0 • a statement that the value of a population parameter is equal to some claimed value • the prediction that there is no interaction between variables

Alternative hypothesis • denoted by H1 or Ha or HA • a statement that the parameter has a value that somehow differs from the null hypothesis • the prediction that there is a measurable interaction between variables

Effect the bio-fertilizer ‘x’ on plant growth

Differences Example đ?‘Żđ?&#x;Ž • đ?‘Żđ?&#x;Ž : đ?? = đ?? đ?&#x;Ž • đ?‘Żđ?&#x;Ž : đ?? ≼ đ?? đ?&#x;Ž • đ?‘Żđ?&#x;Ž : đ?? ≤ đ?? đ?&#x;Ž

Example đ?‘Żđ?‘¨ • đ?‘Żđ?&#x;Ž : đ?? ≠đ?? đ?&#x;Ž • đ?‘Żđ?&#x;? : đ?? < đ?? đ?&#x;Ž • đ?‘Żđ?&#x;? : đ?? > đ?? đ?&#x;Ž

Null Hypothesis: đ??ť0

• Must contain condition of equality: =, , or  • We wish to reject • Assume true until proven otherwise.

Alternative Hypothesis: đ??ťđ??´

• The symbolic form of the alternative hypothesis must use one of these symbols: ď‚š, <, >. • Must be true if H0 is false • ‘opposite’ of Null • We support • We trying to prove by conducting the inferential statistics

Type of alternative hypothesis

One-tailed (directional) • Right-tail • Left-tail

Two-tailed (non-directional)

Differences Directional

Onetailed

• predicts the actual DIRECTION • more precise • relies on previous studies

Twotailed

• predicts an OPEN outcome • very general • no other research has been done

• One-tailed

Non-directional • Two-tailed

Examples One-tailed hypothesis

• Example 1

• The national mean cholesterol level is appproximately 210. • 100 people with high cholesterol levels (over 265) participated in a drug study and were treated with a new drug Cholestyramine. • After treatment the sample mean was 228 and the sample standard deviation was 12. • One question of interest is whether people taking this drug still have a mean cholesterol level that exceeds the national average. • What are the null and alternative hypotheses?

Examples One-tailed hypothesis • đ?‘Żđ?&#x;Ž : đ?? = đ?? đ?&#x;Ž • đ?‘Żđ?‘¨ : đ?? > đ?? đ?&#x;Ž

• Solution đ?‘Żđ?&#x;Ž : đ?? = đ?&#x;?đ?&#x;?đ?&#x;Ž đ?‘Żđ?‘¨ : đ?? > đ?&#x;?đ?&#x;?đ?&#x;Ž

• Population Characteristic: • Ο = Average cholesterol level for all people taking this drug

Examples Two-tailed hypothesis

• Example 2 • We have a medicine that is being manufactured and each pill is supposed to have 14 milligrams of the active ingredient. • What are our null and alternative hypotheses?

Examples Two-tailed hypothesis • đ?‘Żđ?&#x;Ž : đ?? = đ?? đ?&#x;Ž • đ?‘Żđ?‘¨ : đ?? ≠đ?? đ?&#x;Ž

• Solution đ?‘Żđ?&#x;Ž : đ?? = đ?&#x;?đ?&#x;’ đ?‘Żđ?‘¨ : đ?? ≠đ?&#x;?đ?&#x;’ • Our null hypothesis states that the population has a mean equal to 14 milligrams. • Our alternative hypothesis states that the population has a mean that is different than 14 milligrams.

Possible outcomes Reject đ?‘Żđ?&#x;Ž and accept đ?‘Żđ?‘¨ • sufficient evidence in the sample in favor or đ?‘Żđ?‘¨

Do not reject đ?‘Żđ?&#x;Ž • insufficient evidence to support đ?‘Żđ?‘¨

• Select the significance level STEP 2: (α)

Significance level (α) • Typical level • 0.10, 0.05, 0.01 • Make decision • Statistically significant • did not occur by random chance • Not statistically significant • Show how likely the null hypothesis is true • Example, when α=0.10, there is 10% chance of rejecting the true null hypothesis

• Select the significance level STEP 2: (α)

Significance level (α) • Typical level • 0.10, 0.05, 0.01 • Make decision • Statistically significant • did not occur by random chance • Not statistically significant • Show how likely the null hypothesis is true • Example, when α=0.10, there is 10% chance of rejecting the true null hypothesis

STEP 3:

• Identify the test statistic

Test statistic • Population mean • (n>30) • (n<30) • Population proportion

Population mean

Test statistic • z-test • t-test

• The test statistic is the tool

• we use to decide whether or not to reject the null hypothesis.

• It is obtained by taking the observed value (the sample statistic)

• converting it into a standard score under the assumption that the null hypothesis is true.

• Large sample

• Use Z-test • When n ≥ 30, σ is KNOWN

• Small sample

• Use t-test • When n < 30, σ is UNKNOWN

Z-Test statistic

đ?‘Ľâˆ’đ?œ‡ đ?‘Ľâˆ’đ?œ‡ đ?‘§= = đ?œŽ đ?œŽđ?‘Ľ đ?‘› • Where

Population mean (n>30)

• đ?‘Ľ = sample mean • đ?œ‡ = â„Žđ?‘Śđ?‘?đ?‘œđ?‘Ąâ„Žđ?‘’đ?‘ đ?‘–đ?‘ đ?‘’đ?‘‘ đ?‘?đ?‘œđ?‘?đ?‘˘đ?‘™đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› đ?‘šđ?‘’đ?‘Žđ?‘› • Ďƒ = population standard deviation • đ?‘› = đ?‘ đ?‘Žđ?‘šđ?‘?đ?‘™đ?‘’ đ?‘ đ?‘–đ?‘§đ?‘’

T-Test statistic

đ?‘Ľâˆ’đ?œ‡ đ?‘Ľâˆ’đ?œ‡ đ?‘Ą= = đ?‘ đ?‘ đ?‘Ľ đ?‘› • Where

Population mean (n<30)

• đ?‘Ľ = sample mean • đ?œ‡ = â„Žđ?‘Śđ?‘?đ?‘œđ?‘Ąâ„Žđ?‘’đ?‘ đ?‘–đ?‘ đ?‘’đ?‘‘ đ?‘?đ?‘œđ?‘?đ?‘˘đ?‘™đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› đ?‘šđ?‘’đ?‘Žđ?‘› • s = sample standard deviation • đ?‘› = đ?‘ đ?‘Žđ?‘šđ?‘?đ?‘™đ?‘’ đ?‘ đ?‘–đ?‘§đ?‘’

Z-Test statistic

Proportion • Binomial

𝑥−𝜇 𝑥 − 𝑛𝑝 𝑧= 𝑜𝑟 𝑧 = 𝜎 𝑛𝑝𝑞 • Where • 𝜇 = 𝑛𝑝 • 𝜎 = 𝑛𝑝𝑞 • 𝑝 = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 • When • 𝑛𝑝 ≥ 5 𝑎𝑛𝑑 𝑛𝑞 ≤ 5

• Formulate the decision rule STEP 4:

Decision rule • z-test • t-test

Method Critical value

P-value

Differences Critical value

P-value

Critical value approach • A traditional approach • The critical value is the standard score that separates the rejection region (α) from the rest of a given curve.

P-value • The P-value for any given hypothesis test is the probability of getting a sample statistic at least as extreme as the observed value. • That is to say, it is the area to the left or right of the test statistic.

P-value

• Make statistical decision STEP 4:

Statistical decision

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.