The one and two sample proportion hypothesis tests involving one factor with one and two samples, these tests may assumes a binomial distribution. If more than two samples exist then use Chi-Square test.

## One Sample Proportion Hypothesis Test

The One Sample Proportion Test is used to estimate the proportion of a population. It compares the proportion to a target or reference value and also calculates a range of values that is likely to include the population proportion. This is also called hypothesis of inequality.

Use normal approximation or binomial enumeration based on the sample size. If the sample size is large, then normal approximation will give more accurate results. If the sample sizes is less then binomial enumeration gives much more accurate results.

## Assumptions of the one sample Proportion test

- The data are simple random values from the population
- Population follows a binomial distribution
- When both mean (np) and variance( n(1-p)) values are greater than 10, the binomial distribution can be approximated by the normal distribution

## Hypothesis of one sample proportion test

- Null hypothesis: population proportion is equal to hypothesized proportion
- Alternative hypothesis: population proportion is not equal to hypothesized proportion (two -tailed)
- Population proportion is greater than hypothesized proportion (one -tailed)
- Population proportion is less than hypothesized proportion (one -tailed)

## Test statistic for one sample proportions test

Where

- z is test statistic
- p̂is observed proportion
- P
_{0}is hypothesized probability - n is sample size

## Procedure to execute One Sample Proportion Hypothesis Test

- State the null hypothesis and alternative hypothesis
- State alpha, in other words determine the significance level
- Compute the test statistic
- Determine the critical value (from critical value table)
- Define the rejection criteria
- Finally, interpret the result. If the test statistic falls in critical region, reject the null hypothesis

## Example of One Sample Proportion Test

A researcher claims that Republican Party will win in next Senate elections especially in Florida State. A statistical data reported that 23% voted for Republican Party in last election. To test the claim a researcher surveyed 80 people and found 22 said they voted for Republican Party in last election. Is there enough evidence at α=0.05 to support this claim?

- P
_{0}=0.23 - n=80
- p̂=22/80=0.275

Define Null and Alternative hypothesis

- Null Hypothesis: p= 0.23
- Alternative Hypothesis: p≠0.23

State Alpha

- α=0.05

State decision rule

Critical value is ±1.96, hence reject the null hypothesis if the calculated value is less than -1.96 or greater than +1.96

Calculate test statistic

=0.045/0.047= 0.957

Since calculated value is in between -1.96 and 1.96 and it is not in critical region, hence failed to reject the null hypothesis.

## Two Sample Proportion Hypothesis Tests

Two sample proportion test is used to determine whether the proportions of two groups differ. It calculates the range of values that is likely to include the difference between the population proportions.

## Assumptions of the Two Sample Proportion Hypothesis Tests

- The data are simple random values from both the populations
- Both populations are follows a binomial distribution
- When both mean (np) and variance( n(1-p)) values are greater than 10, the binomial distribution can be approximated by the normal distribution

## Hypothesis of two sample proportion test

- Null hypothesis: The difference between population proportions is equal to hypothesized difference, in short p
_{1}= p_{2} - Alternative hypothesis: The difference between population proportions is not equal to hypothesized difference (two -tailed)
- The difference between population proportions is greater than hypothesized difference (one -tailed)
- The difference between population proportions is less than hypothesized difference (one -tailed)

## Test statistic for two sample proportions test

Where

- z is test statistic
- p̂
_{1}_{ }and p̂_{2}are observed proportion of events in the two samples - n is sample size
- X
_{1}and X_{2}are number of trails

## Procedure to execute Two Sample Proportion Hypothesis Test

- State the null hypothesis and alternative hypothesis
- State alpha, in other words determine the significance level
- Compute the test statistic
- Determine the critical value (from critical value table)
- Define the rejection criteria
- Finally, interpret the result. If the test statistic falls in critical region, reject the null hypothesis

## Example of Two Sample Proportion Test

A car manufacturer aims to improve the quality of the products by reducing the defects and also increase the customer satisfaction. Therefore, he monitors the efficiency of two assembly lines in the shop floor. In line A there are 18 defects reported out of 200 samples. While the line B shows 25 defects out of 600 cars. At α 5%, is the differences between two assembly procedures are significant?

Define Null and Alternative hypothesis

- Null Hypothesis: Two proportions are the same
- Alternative Hypothesis: Two proportions are not the same

α=0.05

State decision rule

Critical value is ±1.96, hence reject the null hypothesis if the calculated value is less than -1.96 or greater than +1.96

Calculate Test Statistic

- Line A= p̂
_{1=}18/200= 0.09 = 9% - Line B = p̂
_{2=}25/600 = 0.0416 = 4.16%

p_{0} = 18+25/200+600=43/800=0.0537 =5.37%

=2.62

Calculated test statistic value 2.62 and it is in critical region, hence reject the null hypothesis, so, there is a significant difference in two line assembly procedures

**Six Sigma Black Belt Certification One sample and Two sample Proportion Test Questions:**

**Question 1:** Which of the following statement is true, the right tailed test of a single sample proportion test statistic value is +1.12 and the critical value from the table is +2.89.

(A) Reject the null hypothesis

(B) Failed to reject the null hypothesis

(C) Accept the null alternative hypothesis

(D) None of the above

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**Question 2:** What could be the null hypothesis for a two sample proportion test, if the alternative hypothesis is p_{1} <p_{2}?

(A) Null hypothesis : p_{1} > p_{2}

(B) Null hypothesis : p_{1} = p_{2}

(C) Null hypothesis : p_{1} < p_{2}

(D) Null hypothesis : p_{1} ≠ p_{2}