Null and Alternative Hypothesis

In hypothesis testing, the null hypothesis (denoted by Ho) is a neutral statement, is tested against an Alternative hypothesis (denoted by H1). The conclusion of a hypothesis testing is that I either reject the Null Hypothesis or fail to reject the null hypothesis. If the direction of claim is mentioned it is one tailed hypothesis or directional hypothesis. If the direction of claim is not mentioned it is a two-tailed hypothesis or non-directional hypothesis.

Alpha

Alpha is the Probability of Type I which is also known as level of significance. It’s generally taken as 1%, 5% or 10%. With alpha equal to 5% I am 95% confident about my results. 

Type I and Type II error

In hypothesis testing, there can be 2 types of errors, type I and type II. Type I error is the rejection of null hypothesis when it is actually true. While Type II error is not rejecting the null hypothesis when it is actually false. The probability of Type I error is alpha, while the probability of Type II error is known as beta. Wrongly rejecting a null hypothesis is considered more serious as compared to wrongly accepting it.

Critical Region

The critical region is defined as a region of outcome set where the null hypothesis is rejected. Alpha, the probability of type I error is also known as the size of the critical region. The decision rule is based on p-value approach or critical value approach. The critical value approach is based on the concept that if the value of test statistic lies in between the critical region (W) then I reject the null. Otherwise, if the value of test statistic falls in the complement of the critical region I fail to reject the null. 

P-value

P-value is the probability that null hypothesis is true. The decision rule is based on p-value approach or critical value approach. If the p-value is less than alpha, I reject the null hypothesis at alpha% level of significance. Else if the p-value is greater than alpha, I fail to reject the null hypothesis at alpha% level of significance. With p-value being 0.06, I can say that there is 6% chance that null hypothesis is true. 

Can you prove of disapprove a Null hypothesis?

In hypothesis testing, the null hypothesis is never proved or established, but is possibly disproved. Testing of hypothesis begins with assuming that null hypothesis is true, but it is up to the experimenter to prove it false. If the researcher proves it to be false, then the null hypothesis is rejected. But if the researcher cannot prove that the null hypothesis is false, it does not mean the null will be accepted as true because we had initially set out with the assumption that the null is true. This is why we either reject the null hypothesis or fail to reject the null hypothesis, but never accept it.

Consider the example, null hypothesis Ho: The person is innocent vs. alternative hypothesis Ha: The person is not innocent (is guilty) Now, it is up to the prosecution to build a case to prove guilt beyond a reasonable doubt. It should be noted here that a jury can never prove a person to be innocent. The defendant can only be declared not guilty. i.e., the jury has failed to reject the null hypothesis, but it has not accepted it.

• Assumptions
  

1. One sample Z test is used when the sample size is large (> 30) or population standard deviation is known.
   
2. Method of simple random sampling is used
   
3. Data is normally distributed
   

• 5-Step Hypothesis
  

Null and Alternative Hypothesis

• Ho: population means is equal to μo. μ = μo
• Ha1: population means is equal to μo. μ≠μo  (Two-tailed test)

Ha2: population means is equal to μo. μ<μo  (Left tailed test)

Ha3: population means is equal to μo. μ>μo  (Right tailed test)

Here μ_o takes a particular value.

Test Statistic

Critical value

• -z(a/2), z(a/2) (Two-tailed test)
• -z(a) (Left tailed test)
• z(a) (Right tailed test)

P-value

• 2*(1-P(Z≤|z|)  (Two-tailed test)
• P(Z≤z)  (Left tailed test)
• P(Z≥z)  (Right tailed test)

Decision rule

• Reject Ho if |z|>z(a/2) or p-value < alpha (two-tailed test)
• Reject Ho if –z < -z(a) or p-value < alpha (left tailed test)
• Reject Ho if z > z(a) or p-value < alpha (right-tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population mean is given by:

This implies I am 100(1-alpha)% confident that estimated population mean lies in the obtained interval.

One sample z test CALCULATOR

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• Assumptions

• ·        One sample t test is used when sample size is small (<= 30) or population standard deviation is unknown.
• ·        Method of simple random sampling is used
• ·        Data is normally distributed

• 5 Step Hypothesis

Hypothesis

Ho: population mean is equal to μo. μ = μo

Ha1: population mean is equal to μo. μ≠μo  (Two tailed test)

Ha2: population mean is equal to μo. μ<μo  (Left tailed test)

Ha3: population mean is equal to μo. μ>μo  (Right tailed test)

Here μo takes a particular value.

 Test Statistic

 Critical value

-t(a/2,n-1),t(a/2,n-1) (Two tailed test)

-t(a,n-1) (Left tailed test)

t(a,n-1) (Right tailed test)

P-value

2*(1-P(T≤|t|)  (Two tailed test)

P(T≤t)  (Left tailed test)

P(T≥t)  (Right tailed test)

 Decision rule

Reject Ho if |t| > t(a/2, df) or p-value < alpha (two tailed test)

Reject Ho if –t < -t(a, df) or p-value < alpha (left tailed test)

Reject Ho if t > t(a, df) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

 100(1-alpha)% Confidence interval for population mean is given by:

This implies I am 100(1-alpha)% confident that estimated population mean lies in the obtained interval.

One sample t test CALCULATOR

• Assumptions

Method of simple random sampling is used

There are only two outcomes, p (probability of success) or q (probability of failure). Also, p + q = 1

If np≥10 and nq≥10, then normal approximation to binomial is valid

• 5 step Hypothesis

Hypothesis

Ho: population proportion is equal to po. p = po

Ha1: population mean is equal to po. p≠po  (Two tailed test)

Ha2: population mean is equal to po. p<po  (Left tailed test)

Ha3: population mean is equal to po. p>po  (Right tailed test)

Here p_o takes a particular value.

 Test Statistic

 Critical value

-z(a/2), z(a/2) (Two tailed test)

-z(a) (Left tailed test)

z(a) (Right tailed test)

 P-value

2*(1-P(Z≤|z|)  (Two tailed test)

P(Z≤z)  (Left tailed test)

P(Z≥z)  (Right tailed test)

 Decision rule

Reject Ho if |z|>z(a/2) or p-value < alpha (two tailed test)

Reject Ho if –z < -z(a) or p-value < alpha (left tailed test)

Reject Ho if z > z(a) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for population mean is given by:

This implies I am 100(1-alpha)% confident that estimated population proportion lies in the obtained interval.

One sample z test for Proportions CALCULATOR

• Assumptions

1. Dependent variable is continuous (measured on interval or ratio scale of measurement).
   
2. Independent variable is categorical (grouped into 2 independent samples)
   
3. There is no relationship/dependency between 2 samples
4. Method of simple random sampling is used
5. Data is normally distributed
6. Homogeneity of variance between 2 independent samples. This can be tested using Levene’s Test or F test
   
   

• 5-Step Hypothesis

Null and Alternative Hypothesis

• Ho: there is no significant difference in the population means of 2 groups. μ1-μ2=0

• Ha1: there is significant difference in the population means of 2 groups. μ1-μ2≠0 (Two tailed test)

Ha2: population mean of sample 1 is less than that of sample 2. μ1-μ2<0  (Left tailed test)

Ha3: population mean of sample 1 is greater than that of sample 2. μ1-μ2>0  (Right tailed test)

Test Statistic

Critical value

• -t(a/2,n1+n2-2),t(a/2,n1+n2-2) (Two tailed test)
• -t(a,n1+n2-2) (Left tailed test)
• t(a,n1+n2-2) (Right tailed test)

P-value

• 2*(1-P(T≤|t|)  (Two tailed test)
• P(T≤t)  (Left tailed test)
• P(T≥t)  (Right tailed test)

Decision rule

• Reject Ho if |t| > t(a/2, df) or p-value < alpha (two tailed test)
• Reject Ho if –t < -t(a, df) or p-value < alpha (left tailed test)
• Reject Ho if t > t(a, df) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population mean difference is given by:

This implies I am 100(1-alpha)% confident that estimated population mean difference between two samples lies in the obtained interval. If confidence interval contains 0, I fail to reject null hypothesis ho and conclude that there is no significant difference in the means of two samples.

• Assumptions
  

1. Dependent variable is normally distributed and is continuous (measured on ratio/interval scale of measurement).
   
2. Independent variable is categorical (grouped into 2 dependent samples). Example: before-after

• 5-Step Hypothesis
  

Null and Alternative Hypothesis

• Ho: there is no significant difference in the population means of two groups. μ1-μ2=0

• Ha1: there is significant difference in the population means of two groups. μ1-μ2≠0 (Two tailed test)

Ha2: population mean of sample 1 is less than that of sample 2. μ1-μ2<0  (Left tailed test)

Ha3: population mean of sample 1 is greater than that of sample 2. μ1-μ2>0  (Right tailed test)

Test Statistic

Critical value

• -t(a/2,n-1),t(a/2,n-1) (Two tailed test)
• -t(a,n-1) (Left tailed test)
• t(a,n-1) (Right tailed test)

P-value

• 2*(1-P(T≤|t|)  (Two tailed test)
• P(T≤t)  (Left tailed test)
• P(T≥t)  (Right tailed test)

Decision rule

• Reject Ho if |t| > t(a/2, df) or p-value < alpha (two tailed test)
• Reject Ho if –t < -t(a, df) or p-value < alpha (left tailed test)
• Reject Ho if t > t(a, df) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population mean difference is given by:

This implies I am 100(1-alpha)% confident that estimated population mean difference between two samples lies in the obtained interval. If confidence interval contains 0, I fail to reject null hypothesis ho and conclude that there is no significant difference in the means of two samples.

• Assumptions
  

1. Method of simple random sampling is used
   
2. Two samples are independent
3. There are only two outcomes, p (probability of success) or q (probability of failure). Also, p + q = 1
4. If n1 p1≥10 and n1 q1≥10 for sample 1 and n2 p2≥10 and n2 q2≥10 for sample 2, then normal approximation to binomial is valid

• 5-Step Hypothesis
  

Null and Alternative Hypothesis

• Ho: there is no significant difference in the population proportion of two groups. p1-p2=0

• Ha1: there is significant difference in the population proportion of two groups. p1-p2≠0 (Two tailed test)

Ha2: population mean of sample 1 is less than that of sample 2. μ1-μ2<0  (Left tailed test)

Ha3: population mean of sample 1 is greater than that of sample 2. μ1-μ2>0  (Right tailed test)

Test Statistic

Critical value

• -z(a/2), z(a/2) (Two tailed test)
• -z(a) (Left tailed test)
• z(a) (Right tailed test)

P-value

• 2*(1-P(Z≤|z|)  (Two tailed test)
• P(Z≤z)  (Left tailed test)
• P(Z≥z)  (Right tailed test)

Decision rule

• Reject Ho if |z| > z(a/2) or p-value < alpha (two tailed test)
• Reject Ho if –z < -z(a) or p-value < alpha (left tailed test)
• Reject Ho if z > z(a) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population proportions difference is given by:

This implies I am 100(1-alpha)% confident that estimated population mean difference between two samples lies in the obtained interval. If confidence interval contains 0, I fail to reject null hypothesis ho and conclude that there is no significant difference in the means of two samples.

z test for difference in proportions CALCULATOR

• Assumptions

1. Data is normally distributed.

• 5-Step Hypothesis

Null and Alternative Hypothesis

• Ho: σ=σo
• Ha1: σ≠σo (Two tailed test)

Ha2: σ<σo (Left tailed test)

Ha3: σ>σo (Right tailed test)

Test Statistic

Critical value

• chisq(1-a/2,n-1), chisq(a/2,n-1) (Two tailed test)
• chisq(1-a,n-1) (Left tailed test)
• chisq(a,n-1) (Right tailed test)

P-value

• 2*(1-P(CHISQ≤|Chisq|) (Two tailed test)
• P(CHISQ≤Chisq) (Left tailed test)
• P(CHISQ≥Chisq) (Right tailed test)

Decision rule

• Reject Ho if chisq < chisq(1-a/2,n-1)  or chisq > chisq(a/2,n-1)  or p-value < alpha (two tailed test)
• Reject Ho if chisq < chisq(1-a,n-1)  or p-value < alpha (left tailed test)
• Reject Ho if chisq > chisq(a,n-1)  or p-value < alpha (right tailed test)

• Confidence Interval

100(1-alpha)% Confidence interval for the population variance is given by:

This implies I am 100(1-alpha)% confident that estimated population variance lies in the obtained interval.

• Assumptions

1. Data is normally distributed.

• 5-Step Hypothesis
  

Null and Alternative Hypothesis

• Ho: σ1=σ2
• Ha1: σ1≠σ2 (Two tailed test)

Ha2: σ1<σ2 (Left tailed test)

Ha3: σ1>σ2 (Right tailed test)

Test Statistic

Critical value

• -F(1-a/2,n1-1,n2-1), F(1-a/2,n1-1,n2-1) (Two tailed test)
• F(1-a,n1-1,n2-1) (Left tailed test)
• F(a,n1-1,n2-1) (Right tailed test)

P-value

• 2*(1-P(F≤|f|) (Two tailed test)
• P(F≤f) (Left tailed test)
• P(F≥f) (Right tailed test)

Decision rule

• Reject Ho if F < F(1-a/2,n1-1,n2-1)  or F > F(a/2,n1-1,n2-1)  or p-value < alpha (two tailed test)
• Reject Ho if F < F(1-a,n1-1,n2-1) or p-value < alpha (left tailed test)
• Reject Ho if F > F(a,n1-1,n2-1) or p-value < alpha (right tailed test)

• Confidence Interval

100(1-alpha)% Confidence interval for the population variance is given by:

This implies I am 100(1-alpha)% confident that estimated difference in the population variance of two samples lie in the obtained interval.

• Assumptions

• Method of simple random sampling is used

• Variable is measured on nominal scale of measurement

• Expected value of each of cell is at least 5

• 5 Step Hypothesis

• Hypothesis

• Ho: data follows a specific distribution/proportion. That is p1=p2=p3=…=pi ; i=1,2,3,…,n
• Ha: at least one of the given proportions give above is not true. p1≠ p2≠ p3≠…≠pi ; i=1,2,3,…,n

• Test Statistic

• Critical value

• chisq (a,df)  

• P-value

• P(CHISQ≥Chisq)  

• Decision rule

• Reject Ho if chisq > chisq(a,df)  or p-value < alpha

• Assumptions

• Method of simple random sampling is used
• variable is measured on nominal scale of measurement
• expected value of each of cell is at least 5

• 5 Step Hypothesis

• Hypothesis

Ho: each population has the same proportion of observations. That is

P(level_1) population 1 = P(level_1) population 2 = . . . = P(level_1) population r

P(level_2) population 1 = P(level_2) population 2 = . . . = P(level_2) population r

.

.

P(level_c) population 1 = P(level_c) population 2 = . . . = P(level_c) population r

 

Ha: at least one of population has the different proportion of observations. That is

P(level_1) population 1 ≠ P(level_1) population 2 ≠ . . . ≠ P(level_1) population r

P(level_2) population 1 ≠ P(level_2) population 2 ≠ . . . ≠ P(level_2) population r

.

.

P(level_c) population 1 ≠ P(level_c) population 2 ≠ . . . ≠ P(level_c) population r

• Test Statistic

• Critical value

chisq(a,df)  

• P-value

P(CHISQ≥Chisq)  

• Decision rule

Reject Ho if chisq > chisq(a,df)  or p-value < alpha

• Assumptions

• Method of simple random sampling is used
• variable is measured on nominal scale of measurement
• expected value of each of cell is at least 5

• 5 Step Hypothesis

• Hypothesis

Ho: each population has the same proportion of observations. That is

P(level_1) population 1 = P(level_1) population 2 = . . . = P(level_1) population r

P(level_2) population 1 = P(level_2) population 2 = . . . = P(level_2) population r

.

.

P(level_c) population 1 = P(level_c) population 2 = . . . = P(level_c) population r

 

Ha: at least one of population has the different proportion of observations. That is

P(level_1) population 1 ≠ P(level_1) population 2 ≠ . . . ≠ P(level_1) population r

P(level_2) population 1 ≠ P(level_2) population 2 ≠ . . . ≠ P(level_2) population r

.

.

P(level_c) population 1 ≠ P(level_c) population 2 ≠ . . . ≠ P(level_c) population r

• Test Statistic

• Critical value

chisq(a,df)  

• P-value

P(CHISQ≥Chisq)  

• Decision rule

Reject Ho if chisq > chisq(a,df)  or p-value < alpha

• Between-participants

One of the main advantages of this group is that it avoids practice and fatigue effect as people are restricted to one condition only. While, disadvantage is that a large number of participants is required. Individual differences can give rise to high variability in the moods. Researcher does not have much control over confounding variables.

• Within-participants

One of the main advantages is that fewer people are required. Disadvantages include order effects. That is performance on second condition might be affected cause of practice or fatigue effect.

• Matched pairs

Main advantage is that it avoids order effects and hence counter balancing is not necessary and it requires less participants. While disadvantage is that it required a lot of time to match pairs. It is also impossible to match people exactly.

• Assumptions

1. Homogeneity of variance
2. Dependent variable is measured on ratio/interval scale of measurement
3. Independent variable is measured on nominal scale of measurement
4. Independent variable has at least 3 categories

• 5 Step Hypothesis
  

• Hypothesis

Ho: there is no significant difference in the means of categories of independent variable. That is μ1=μ2=…=μn

Ha: at least one of the means of categories of independent variable differs significantly. That is μ1≠μ2≠…≠μn

• ANOVA Table

• Critical value

F(1-a/2,n_1-1,n_2-1), F(a/2,n_1-1,n_2-1)  

• P-value

2*(1-P(F≤|f|))  

• Decision rule

Reject Ho if F < F(1-a/2,n_1-1,n_2-1)  or F > F(a/2,n_1-1,n_2-1)

• Post Hoc Analysis
  

To determine which of the means are significantly different from one another Post Hoc Analysis is used. Simultaneous comparisons between all pairs of groups can be done here. Turkey Crammer, Bonferroni procedure, Scheffe’s method, Dunnett’s correction are methods of Post Hoc analysis.

The formula for critical range of Turkey Crammer is given by following formula.

A specific pair is said to be significantly different if the absolute difference is greater than the critical range.

• Definition of Two way ANOVA with Replication

A replication study involves repeating a study using the same methods but with different subjects and experimenters. Main focus of two way ANOVA with replication is test for significance of interaction between treatments.

For example I want to test each of three blends of fertilizers on one sample of each of four types of crops. Each combination of blend and crop has a sample of size 5. Here I consider three set of hypothesis statement given below.

Null Hypothesis, Ho1: there is no significant difference between mean of three blends. V/s Alternative Hypothesis, H11: there is significant difference between mean of three blends.

Null Hypothesis, Ho2: there is no significant difference between mean of four types of crops. V/s Alternative Hypothesis, H12: there is significant difference between mean of four types of crops.

Null Hypothesis, Ho3: there is no significant interaction effect between blends and type of crop. V/s Alternative Hypothesis, H13: there is significant interaction effect between blends and type of crop.

• 5 Step Hypothesis
  

• Hypothesis

Null Hypothesis, Ho1: there is no significant difference between mean of levels of treatment A. v/s Alternative Hypothesis, H11: there is significant difference between mean of levels of treatment A.

Null Hypothesis, Ho2: there is no significant difference between mean of levels of treatment B. v/s Alternative Hypothesis, H12: there is significant difference between mean of levels of treatment B.

Null Hypothesis, Ho3: there is no significant interaction effect between treatment A and B. V/s Alternative Hypothesis, H13: there is significant interaction effect between treatment A and B.

• ANOVA Table

• Critical value

F(1-a/2,n1-1,n2-1), F(a/2,n1-1,n2-1)  

• P-value

2*(1-P(F≤|f|))  

• Decision rule

Reject Ho if F < F(1-a/2,n1-1,n2-1)  or F > F(a/2,n1-1,n2-1)

• Definition of Two way ANOVA with Replication

Null Hypothesis, Ho2: there is no significant difference between mean of four types of crops. V/s Alternative Hypothesis, H12: there is significant difference between mean of four types of crops.

• 5 Step Hypothesis
  

Hypothesis:
Null Hypothesis, Ho1: there is no significant difference between mean of levels of treatment A. v/s Alternative Hypothesis, H11: there is significant difference between mean of levels of treatment A.
Null Hypothesis, Ho2: there is no significant difference between mean of levels of treatment B. v/s Alternative Hypothesis, H12: there is significant difference between mean of levels of treatment B.
ANOVA Table:

Critical value: F(1-a/2,n1-1,n2-1), F(a/2,n1-1,n2-1)  

P-value: 2*(1-P(F≤|f|))  

Decision rule: Reject Ho if F < F(1-a/2,n1-1,n2-1)  or F > F(a/2,n1-1,n2-1)

• Assumptions

• Dependent variable and covariate variable(s) should be measured on a continuous scale
• Independent variable should consist of two or more categories
• Independence of observations
• No significant outliers
• Residuals should be approximately normally distributed for each category of the independent variable
• Homogeneity of variances
• Covariate should be linearly related to the dependent variable at each level of the independent variable.

• Hypothesis
  

Ho: main effect of independent variable_i is not significant.

ha: main effect of independent variable_i is significant.

• Critical value
  

F(1-a/2,n-1,n2-1), F(a/2,n1-1,n2-1)  

• P-value
  

2*(1-P(F≤|f|))  

• Decision rule
  

Reject Ho if F < F(1-a/2,n1-1,n2-1)  or F > F(a/2,n1-1,n2-1)

Non parametric test includes, Wilcoxon rank-sum test, Wald Wolfowitz runs test, Mann-Whitney U test, and Kolmogorov-Smirnov, Kruskal Wallis analysis of ranks and the Median test etc.

• Wilcoxon rank sum test for paired samples is counterpart of the paired samples t-test. 
• Wilcoxon signed rank test is equivalent to one-sample t test. 
• Wald Wolfowitz runs test, Mann Whitney U test, and Kolmogorov Smirnov two sample test. 
• The Friedman test is a non-parametric test for testing the difference between several related samples. 
• The Friedman test is an alternative for Repeated measures ANOVA which is used when the same parameter has been measured under different conditions on the same subjects. 
• Kruskal Wallis analysis of ranks and the Median test is non parametric version of one way ANOVA.