Two Sample Z Hypothesis Tests is a parametric test is to compare the means of two independent groups of samples drawn from a normal population. In other words, it is the test to know the means of two populations are the same or different. For example, the mean salaries of male workers are greater than the female workers for the same job?
When to use Two sample Z hypothesis tests
Two sample Z test compares the means of samples of independent groups taken from a normal population. Also, in this test you will have only two groups with independent samples. To compare three groups, use One Way ANOVA instead of Z test.
Furthermore, Z test is similar to the student t-test. Z test is basically used for relatively large samples (say n>30) and the population standard deviation is known. Whereas, student t-test is for small sample size, and also t-test assumes the population standard deviation is unknown.
Difference between One and Two sample Z hypothesis test
The two-sample z test is to tests the difference between means of two groups, whereas a one-sample z test is to tests the difference between a single group and the hypothesized population value.
Similar spread between the groups, in other words homogeneity of variance
Both the samples should be randomly selected from the population
Two sample Z-test Formula
x̅1 = sample mean of first sample
x̅2 = sample mean of second sample
µ1= Mean of first population
µ2= Mean of second population
σ12= population variance in first population
σ22= population variance in second population
n1= sample size of first group
n2= sample size of second group
Hypothesis Testing
A tailed hypothesis is an assumption about a population parameter. The assumption may or may not be true. A one-tailed hypothesis is a test of hypothesis where the area of rejection is only in one direction. Whereas two-tailed, the area of rejection is in two directions. The selection of one or two-tailed tests depends upon the problem.
This page uses a cumulative z table: each value is the area to the left of z, which is why we look up 0.95 to get 1.645 and 0.975 to get 1.96. If your handbook’s table shows the area from the mean (0 to z), look up 0.45 or 0.475 instead; if it shows the upper tail, look up 0.05 or 0.025.
Select appropriate statistic- one-tailed or two-tailed?
Determine the null hypothesis and alternative hypothesis
Determine the level of significance
Find the critical value
Calculate the standardized test statistics
Then make a decision to reject or fail to reject the null hypothesis. Reject the null hypothesis, If the test statistic falls in the critical region.
Finally, interpret the decision in the context of the original claim.
Example of Right-tailed test
Example: Princeton school science teacher claims that students in his section will score higher marks than those in his colleague’s section. The mean science score for 60 students in his section is 22.1, and the standard deviation is 4.8. The mean science score for 40 of the colleagues’ sections is 18.8, and the standard deviation is 8.1. At α = 0.05, can the teacher’s claim be supported?
State the null and alternative hypothesis
Null hypothesis H0: µ1 ≤ µ2
Alternative hypothesis H1 : µ1 > µ2
Select appropriate statistic- Since the claim is student scores higher marks than the colleagues section; it is a right-tailed test.
Level of significance: α= 0.05
Find the critical value: 1-α= 1-0.05=0.95
Look at the 0.95 in z table= 1.645
Calculate the test statistics
x̅1 = 22.1
x̅2 = 18.8
σ1= 4.8
σ2= 8.1
n1= 60
n2= 40
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. zcal value is in the rejection region. Hence, we have enough evidence to reject the null hypothesis. So, at 5%, we have enough evidence to support the teacher’s claim that his students score higher than his colleague students.
Example: A dietician claims that participants in the X program yield less weight reduction than the Y program. The mean weight reduction for 100 participants in the X program is 19.7lb, and the standard deviation is 6.2. Similarly, for program Y, for 120 participants, it is 21.2lb, and the standard deviation is 7.6. At 1% level of significance, can you support the dietician claim?
State the null and alternative hypothesis
Null hypothesis H0: µ1 ≥ µ2
Alternative hypothesis H1 : µ1 < µ2
Select appropriate statistic- Since the claim is participants in X program yields less weight reduction than Y program; it is a left-tailed test.
Level of significance: α= 0.01
Find the critical value: 1-α= 1-0.01=0.99
Look at the 0.99 in z table= 2.330
Calculate the test statistics
x̅1 = 19.7
x̅2 = 21.2
σ1= 6.2
σ2= 7.6
n1= 100
n2= 120
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. zcal value is not in the rejection region. Hence, we failed to reject the null hypothesis. So, at 1% we don’t have enough evidence to support the dietician’s claim.
Example: Glenbrook sports center is planning to compare the ages from a random sample of male and female swimmers in their coaching center. The swimming coordinator collected 62 female swimmers’ data, and the mean age is 23.1 with a standard deviation of 3.5. Similarly, collected 46 male swimmers data and the mean average is 19.2 with a standard deviation of 4.8. Assume the population follows a standard normal distribution. At 5% significance level, test whether there is a significant difference in age between the sexes?
State the null and alternative hypothesis
Null hypothesis H0: µ1 = µ2
Alternative hypothesis H1 : µ1 ≠ µ2
Select appropriate statistic- Since the Glenbrook sports center planning to test whether there is a significant difference in age between the sexes ; it is a two-tailed test.
Level of significance: α= 0.05
Since it is a two tailed test α/2= 0.05/2=0.025
Find the critical value: 1-α/2= 1-0.025=0.975
Then, look at the 0.975 in z table= ±1.96
Calculate the test statistics
x̅1 = 23.1
x̅2 = 19.2
σ1= 3.5
σ2= 4.8
n1= 62
n2= 46
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. zcal value is in the rejection region. Hence, we have enough evidence to reject the null hypothesis. So, at 5% we have enough evidence to support the significant difference in age between the sexes.
Single sample test of a given proportion is used to estimate the proportion of the population. For example, to estimate the proportion of Spanish-speaking students in a school is lower than the state proportion. Basically, it is used to compare the proportion to a target or reference value.
Furthermore, the proportion test is to determine the percentage of individuals who have a particular characteristic. It also calculates a range that is likely to include the population proportion. Hence, proportion involves a binomial distribution.
Single sample test of a given proportion Formula
Where
Z is test statistic
p̂ is the sample or observed proportion
P0 hypothesized proportion
n sample size
When to use
We use the Single sample test of a given proportion when both the independent (X) and dependent variables are discrete. Hence it follows a binomial distribution.
Assumptions of Single sample test of a given proportion test
Population follows a binomial distribution.
Sample is unbiased and representative
Both main and variance np0, n(1- p0) for binomial distribution are both ≥5, then the sampling distribution can be approximated by the normal distribution.
Hypothesis of Single sample test of a given proportion
Null hypothesis H0: population proportion is equal to hypothesized proportion, in other words, p=p0
Alternative hypothesis H1: population proportion is not equal to hypothesized proportion p≠p0 (Two-tailed)
H1: Population proportion is less than hypothesized proportion p<p0 (One-tailed)
H1: Population proportion is greater than hypothesized proportion p>p0 (One-tailed)
How to calculate Single sample test of a given proportion
First, estimate the proportion p, as p̂ =x/n
Where x is the sample who have same characteristic, n is the sample size
Select appropriate statistic- one-tailed or two-tailed?
State the null hypothesis and alternative hypothesis
State alpha, in other words determine the significance level
Define the rejection criteria
Check the assumption, both np0, and n(1- p0)≥5
Compute the test statistic,
Determine z critical value
Finally, interpret the result. If the test statistic falls in critical region, then reject the null hypothesis.
Example of Single sample test of a given proportion
Right-Tailed
Example: A property leasing office is conducting a vaccination drive, and they are encouraging more women to participate in the event. While a sample of 4300 vaccination reports, it is indicated that 2200 are women. Property manager is confident that at least 49 percent of women participated in the drive. At α=0.05, were their vaccination drive efforts successful?
P0=0.49
x=2200
n=4300
p̂=x/n=2200/4300=0.512
Define Null and Alternative hypothesis
Null Hypothesis: population proportion is equal to hypothesized proportion p= 0.49
Alternative Hypothesis: population proportion is greater than hypothesized proportion p>0.49
State Alpha
α=0.05
Both np0, and n(1- p0)≥5
Calculate test statistic
This page uses a cumulative z table: each value is the area to the left of z (z = 2.89 gives 0.9981). If your handbook’s table shows the area from the mean (0 to z), add 0.5 to its value for positive z to match; if it shows the upper tail, subtract its value from 1.
Since, we are looking for longer, P(Z>2.89) = 1- P(Z<2.89)= 1-0.9981 = 0.0019
Since p value (0.0019) is less than the alpha value (0.05), we have enough evidence to reject the null hypothesis which makes our test significant. Therefore vaccination drive is successful
Example: A plastic basket company produces articles with a 30% defective rate. A wholesale supplier of new material claims that the use of their material reduces the defective rate. A random sample of 200 baskets was tested, and 50 were defective units. Now test at 5% significance level, can the supplier claim be accepted?
P0=0.30
x=50
n=200
p̂=x/n=50/200=0.25
Define Null and Alternative hypothesis
Null Hypothesis: population proportion is equal to hypothesized proportion p= 0.30
Alternative Hypothesis: population proportion is less than hypothesized proportion p<0.30
State Alpha
α=0.05
Both np0, and n(1- p0)≥5
Calculate test statistic
Since we are looking for less than, P(Z<-1.543) = 0.0618
Since p value (0.0618) is greater than the alpha value (0.05), we failed to reject the null hypothesis. Hence the supplier claim cannot be accepted.
Example: Statistical data reported that 23% of doctors recommended Bengay pain relieving cream. A researcher surveyed 80 doctors to test the claim. 22 indicate that they suggest Bengay pain relieving cream. So, is there enough evidence at α=0.05 to support this claim?
P0=0.23
x=22
n=80
p̂=x/n=22/80=0.275
Define Null and Alternative hypothesis
Null Hypothesis: population proportion is equal to hypothesized proportion p= 0.23
Alternative Hypothesis: population proportion is not equal to hypothesized proportion p≠0.23
State Alpha
α=0.05
Both np0, and n(1- p0)≥5
State decision rule
Since it is a two tailed test α/2= 0.05/2=0.025
Find the critical value: 1-α/2= 1-0.025=0.975
Then, look at the 0.975 in z table= ±1.96
The 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
Since the calculated value is between -1.96 and 1.96 and it is not in the critical region, hence failed to reject the null hypothesis.
One sample Z hypothesis test is one of the basic tests of inferential statistics. One sample Z test is a parametric procedure for hypothesis testing. It tests whether the sample mean is significantly different (greater than, less than or not equal) than a population mean when the population’s standard deviation is known. Since we use standard normal distribution to compute the critical values, we often call it a one sample Z test.
When to use one sample Z test
One sample Z test is a robust hypothesis test for violations of standard normal distribution. Z test is similar to student t-test, z test is basically used for relatively large samples (say n>30) and the population standard deviation is known. Whereas, student t-test is for a small sample size, and also t-test assumes the population standard deviation is unknown.
The sample should be randomly selected from the population
One sample Z-test Formula
x̅ = sample mean
µ= population mean
σ= population standard deviation
n= sample size
Steps to Calculate One Sample Z hypothesis test
This page uses a cumulative z table: each value is the area to the left of z, which is why we look up 0.95 to get 1.645 and 0.975 to get 1.96. If your handbook’s table shows the area from the mean (0 to z), look up 0.45 or 0.475 instead; if it shows the upper tail, look up 0.05 or 0.025.
Select appropriate statistic- one-tailed or two-tailed?
Determine the null hypothesis and alternative hypothesis
Determine the level of significance
Find the critical value
Calculate the test statistics
Where
x̅ is observed sample mean
μis population mean
σ is population standard deviation
n is sample size
Then make a decision, reject the null hypothesis; If the test statistic falls in the critical region.
Finally, interpret the decision in the context of the original claim.
Hypothesis Testing
A tailed hypothesis is an assumption about a population parameter. The assumption may or may not be true. One-tailed hypothesis is a test of hypothesis where the area of rejection is only in one direction. Whereas two-tailed, the area of rejection is in two directions. The selection of one or two-tailed tests depends upon the problem.
One Sample z test mostly performed in Analyze phase of DMAIC to check the sample mean is significantly different than a population mean when the population’s standard deviation is known.
Example of right-tailed test
Example: A coaching institute claims that the students’ mean scores in their institute are greater than the 82 marks with a standard deviation of 20. A sample of 81 students is selected, and the mean score is 90 marks. At 95% confidence level, is there enough evidence to support the claim?
State the null and alternative hypothesis
Null hypothesis H0: µ ≤ 82
Alternative hypothesis H1 : µ>82
Select appropriate statistic- Since the claim is student marks are greater than 82 marks; it is a right-tailed test.
Level of significance: α= 0.05
Find the critical value: 1-α= 1-0.05=0.95
Look at the 0.95 in z table= 1.645
Calculate the test statistics
x̅ =90
μ=82
σ =20
n =81
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. Since zcal value is greater than z critical value and it is in the rejection region. Hence we can reject the null hypothesis.
There is enough evidence to support the students’ scores in their institute is greater than the 82 marks.
Example: An online grocery shop claims that the mean delivery time is less than 120 minutes with a standard deviation of 30 minutes. A random sample of 49 orders is delivered with a mean of 100 minutes. At 95% confidence level, is there enough evidence to support the claim?
State the null and alternative hypothesis
Null hypothesis H0: µ ≥ 120
Alternative hypothesis H1 : µ<120
Select appropriate statistic- Since the claim indicates delivery time is less than 120 minutes, it is a left-tailed test.
Level of significance: α= 0.05
Find the critical value: 1-α= 1-0.05=0.95
Look at the 0.95 in z table= -1.645
Calculate the test statistics
x̅ =100
μ=120
σ =30
n =49
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. Since zcal value is less than z critical value and it is in the rejection region. Hence we can reject the null hypothesis.
There is enough evidence to support the claim the mean delivery time is less than 120 minutes.
Example: The mean weight of the rubber gasket is 60 grams with a standard deviation of 48. Furthermore, A researcher thinks that material X will have a significant impact on the weight of a gasket. A random sample of 64 gaskets collected and the mean weight is 68 grams. At 95% confidence level test the material X had an effect.
State the null and alternative hypothesis
Null hypothesis H0: µ = 60
Alternative hypothesis H1 : µ≠60
Since it is a two tailed test α/2= 0.05/2=0.025
Find the critical value: 1-α/2= 1-0.025=0.975
Then, look at the 0.975 in z table= ±1.96
Calculate the test statistics
x̅ =68
μ=60
σ =48
n =64
Interpret the results: Compare Z calc to Z critical . In hypothesis testing, a critical value is a point on the test distribution compares to the test statistic to determine whether to reject the null hypothesis. Since zcal value is not in the rejection region. Hence, we fail to reject the null hypothesis. We can say the mean weight of the gasket is 60.
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One of the biggest challenges Six Sigma students encounter is Hypothesis Testing. There are so many types! And there are so many different equations. How on earth are you supposed to be able to make sense of those problems, especially if you haven’t done word problems in years or ‘you’re just no good at math?’ Not to worry. I’ve got you covered. Here, I’ve put together the Hypothesis Test Study Guide.
Firstly, I aim to provide a cohesive overview of the types of hypothesis tests you’ll be responsible for on your Six Sigma exam, which hypothesis test to use under which circumstances, and an outline of how to do them.
First, a Word on Hypothesis Tests
These concepts fall under the phrase ‘simple but not easy.’
They’re simple because they are easily repeatable algorithms, or a series of instructions, you can repeatedly follow to get the desired result.
The not-easy part is that you must learn all the different types and decide when to use each.
If you can bake cookies, you can do hypothesis tests following this study guide.
Make fun of me all you want. My mental model for learning hypothesis tests for Six Sigma certification was baking.
Instead of thinking of hypothesis testing, consider baking. What if they asked you when to bake cookies instead of cake? Or bake bread instead of pie? And then imagine they were asking you questions about how to bake that pie, cookie, cake, or bread loaf.
There’s a time and a place for each of those baked goods. And also, there’s a time and a place for each hypothesis test.
All you have to do is decide what kind of baked good you want to cook and then pick the right recipe for the given situation!
Let’s dive in by learning about the different types covered in this Hypothesis Test Study Guide.
Furthermore, below, I give you a guided tour of the different types of hypothesis tests: baked goods. I introduce the variations of each type and when you’d want to ‘bake’ that item. I then link to other articles on this website for the ‘recipe’ and examples.
Hypothesis Testing Basics
Since this is a long article, I’ve added additional resources that can help provide an excellent foundation for the Hypothesis Tests Study Guide.
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Okay, let’s dive into how you can pick which hypothesis test to use and when.
What You Need to Know about the Basics of Hypothesis Testing for Your Six Sigma Certification Exam
Hypothesis testing Basics Define and distinguish between statistical and practical significance and apply tests for significance level, power, type I and type II errors. Also, determine appropriate sample size for various test. (Apply).
1. Terminology Define and interpret the significance level, power, type I and type II errors of statistical tests. (Evaluate)
2. Statistical vs. practical significance Define, compare and interpret statistical and practical significance. (Evaluate)
3. Sample size Calculate sample size for common hypothesis tests (e.g., equality of means, equality of proportions, etc.). (Apply)
4. Point and interval estimates Define and distinguish between confidence and prediction intervals. Define and interpret the efficiency and bias of estimators. Calculate tolerance and confidence intervals. (Evaluate)
Define and interpret the efficiency and bias of estimators; compute, interpret and draw conclusions from statistics such as standard error, tolerance intervals, and confidence intervals; understand the distinction between confidence intervals and prediction intervals.
The Three Types of Hypothesis Tests
You can divide the types of hypothesis tests you need into three main types:
Tests for means, tests for variances, and tests for proportions
If this sounds intimidating, think of cookies, cake, and pie! ;’)
Let’s step through each type and how to select each one.
Remember Mean, Median, and Mode? Furthermore, those are fancy statistical words for basic math principles. Mean is a synonym for average. So you might infer that a hypothesis test for means involves testing averages, and you’d be right!
When do you use Hypothesis Tests for Means?
You use a hypothesis test for means when you want to see if two sample populations have a characteristic that is, on average, the same.
A good example is when you need to know if you compare similar samples. For instance, if you needed to see if two samples were, on average, roughly equivalent before proceeding to use them in some manner.
Or you might want to see if the sample you just took is actually reflective of the whole population. For instance, if your sample is just the honors students, that sample is unlikely representative of the entire school.
Why should we run a complicated means test when we could look at the sample averages?
Why do we even need hypothesis tests? If we took a random sample and our sample mean of x differs from y, there’s a difference, right?
Sadly, it’s not that easy because we’re looking at a sample rather than the entire population. We need to account for sampling errors. We also want to know the statistical likelihood of a hypothesis at a given rate of certainty. Comparing the averages is a guess. Sometimes, you can get away with that. But if you want certainty, you will want to use a statistical test.
Note on sampling error:
Sampling error is the difference between a sample and the entire population. Thanks to sampling error, it’s entirely possible that while our sample mean is 330.6, the population mean could still be 260. Or, to put it another way, if we repeated the experiment, it’s possible that the second sample mean could be close to 260. A hypothesis test helps assess the likelihood of this possibility!
Sometimes, you have to interpret the question yourself. Other times, it will be given to you. Sometimes, you can use the details in Step 2 to figure out what your hypotheses should be.
Remember, your null and alternative hypotheses must be mutually exclusive!
Step 2: Pick a type of means test!
There are several different types of means tests. So, be sure to pick the right one for the right job.
Z Test (Is the sample mean the same as the population mean?)
You use a Z test to compare if your sample mean is the same as the population mean. Use it to validate you have a good sample.
Like many hypothesis tests we cover here, you must have normal data.
Furthermore, the Z test is similar to the student t-test. Z test is used for relatively large samples (say n>30), and the population standard deviation is known. Whereas the student t-test is for small sample size, the t-test assumes the population standard deviation is unknown.
There are several kinds of T-tests. You’ll have to decide on each one. The names of each are pretty descriptive. Below, I’ve added common indicators of when to use each.
One Sample T-Test
Use the One-Sample T-Test when you want to know if your sample’s mean is different from another one – either from your imagination or a specification. Or use it when you need to test the mean of a single group against a known mean.
Where
x̅ is observed sample mean
μ0 is the population mean
s is the sample standard deviation
n is the number of observations in the sample
The above formula is a ratio. Also, the resemblance of the t value is the signal-to-noise ratio.
Signal
The Numerator in the above formal is the signal. The Signal is also known as effect size. It is the difference between the sample mean and the null hypothesis. For example, if the sample mean is 12, and the null hypothesis is 9, the signal is 3. If the difference increases, either positively or negatively, it means the strength of the signal increases.
Noise
The Denominator in the above formal is the noise. It measures the standard error of the mean. Noise tells the accuracy of sample estimates of the population mean.
Signal to Noise ratio
The ratio of signal to noise indicates the t value. If the signal is four and the noise is 2, then the t value is 2, which means two times the size of the standard error. If you have a high signal value and low noise, then it produces larger t-values.
A one-sample t-test tests whether a population mean differs significantly from some hypothesized value. Each makes a statement about how the actual population means μ relates to some hypothesized value M.
Two sample T-Test
You guessed it! This one is for two different samples.
A two-sample t-testtests the difference between the two population means. Also, it is a common application to determine whether the means are equal.
Each makes a statement about the difference d0 between the mean of one population μ1 and the mean of another population μ2.
How is the two-sample t-test different from a Paired T?
When would you use a two-sample t-test and not a Paired T?
Paired Sample T-Test
The paired sample t-test, sometimes called the dependent sample t-test, is a statistical procedure used to determine whether the mean difference between two observations is zero. In a paired sample t-test, each subject or entity is measured twice, resulting in pairs of observations.
Use the Paired Sample T-test to determine whether the means of 2 sets are the same or if you must compare means from the same sample group taken at different times.
Good questions to ask yourself:
Why not just take the average value of each set and compare them against each other?
When to use a paired t-test and one sample t-test
A paired t-test compares the mean and standard deviation of two matched groups to determine if there is a significant difference between the two groups. Whereas the One-Sample T Hypothesis Test allows you to compare the (small) population mean to some hypothesized value or one sample, mean to determine if they are significantly different.
One-Way ANOVA
The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.
The null hypothesis for ANOVA is that the mean (average value of the dependent variable) is the same for all groups. The alternative or research hypothesis is that the average is different for all groups. The ANOVA test procedure produces an F-statistic, which calculates the p-value.
We must first understand, ‘What in the world is a variance?’ The root of the word is variability. In other words, in a statistical sense, we’re thinking about how extensive a range of options is in a sample or population.
Consider a standard bell curve graph to get a graphical sense of this. The longer it is, meaning the further it stretches across the y-axis, the more variability it has. “Spread” is another word that refers to variance – as in how far the range of possible values in a given group “spread” across.
Another way to think about Variance questions is to automatically replace the word ” Variance ” with ” Standard Deviation. “
A hypothesis test for proportion is a way to see if there is a difference between two groups. In other words, between a group and a given standard.
When do you use Hypothesis Tests for Proportions?
You would use a Hypothesis test for proportion when you’re asked about the difference between two variables.
What is Needed to Run a Hypothesis Test for Proportions?
Simple, random sampling
Just (2) possible outcomes.
Label 1 outcome a success, the other a failure.
Have at least 10 successes and 10 failures
Finally, the population is 20 times as large as the sample size.
How to do a Hypothesis Test for Proportions
Step 1: Write the Hypothesis and decide which type of tailed test you need.
We’ve covered hypothesis writing earlier. It’s very easy when writing Proportion tests.
Just remember that the null hypothesis ALWAYS includes some kind of equals. (Ex =, <=, or >=)
Furthermore, an alternative hypothesis is a statement that there is a relationship between two variables, a difference between two groups, or a difference from a previous or existing standard.
Let’s move on to the kind of tailed test.
One tail
If you want to reject if something is too small or too big, you’re doing a one-tail test.
But which tail?
Left tail vs. Right tail
Left tail: alternative hypothesis expresses ‘less than.’
Right tail: alternative hypothesis expresses “greater than”
Two tail
For example, If the alternative hypothesis has l=
Step 2: Compute the Standard Deviation (of the sample!)
σ = sqrt[ P *(1 – P)/n]
where P is the hypothesized value of population proportion in the null hypothesis, and n is the sample size.
Step 3: Compute the test statistic
The test statistic is a z-score (z) defined by the following equation.
z = (p – P)/σ
where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and σ is the standard deviation of the sampling distribution.
Step 4: Find the P value
The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a z-score, use that process to assess the probability associated with the z-score.
Step 5: Interpret your results
Given the null hypothesis, the researcher rejects the null hypothesis if the sample findings are unlikely. Typically, this involves comparing the P-value to the significance level and rejecting the null hypothesis when the P-value is less than the significance level.
Additional Helpful Hypothesis Testing Study Guide Videos
Introduction to Hypothesis Testing
How to Choose Which Hypothesis Test to Use
Beyond Normal: Non-Parametric Hypothesis Tests
Finally, Black belt candidates are widely expected to be conversant in non-parametric hypothesis testing across all three types; Means, Variance, and Proportion.
When you’re ready, there are a few ways I can help:
First, join 12,000+ other Six Sigma professionals by subscribing to my email newsletter. A short read every Monday to start your work week off correctly. Always free.
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You’ve spent so much effort learning Lean Six Sigma. Why leave passing your certification exam up to chance? This comprehensive study guide offers 1,000+ exam-like questions for Green Belts (2,000+ for Black Belts) with full answer walkthroughs, access to instructors, detailed study material, and more.
Question: A correlation analysis is used to provide a numeric value for which of the following types of relationships between two variables?
A) Random
B) Linear
C) Curvilinear
D) Causation
Answer: B – Linear. This is a definition of correlation analysis. You could easily rule out random, though – there is no relationship between random events. It would also be helpful to see the articles on correlation coefficient and regression analysis to see this in action.
Question: Which of the following terms is used to describe the risk of a type I error in a hypothesis test?
(A) Power (B) Confidence level (C) Level of significance (D) Beta risk
Answer: (C) Level of significance. The null hypothesis is rejected if the p-value is less than the significance or α level. The α level is the probability of rejecting the null hypothesis given that it is true (type I error) and is most often set at 0.05 (5%).
Question: Which of the following is a commonly accepted level for alpha risk?
(A) 0.05 (B) 0.50 (C) 0.70 (D) 0.95
Answer: (a) 0.05 – Remember that alpha risk is the risk that we have incorrectly rejected the null hypothesis even though it is correct, one of the kinds of errors we can make. 0.05 alpha means that we are comfortable with a 5% chance of incorrectly rejecting the null. That is a common industry standard. Even if you didn’t know that, the other answers don’t make sense. At 50% you are basically flipping a coin! Also see the guide to hypothesis testing.
Question: In an “X” Sifting exercises a Belt will use a(n) _______________ to assist in isolating families of variation that may exist within a subgroup, between subgroups or vary over time.
(A) Multi – Vari Chart (B) Pareto Chart (C) FMEA (D) Shewhart Analysis
Question: When conducting Hypothesis Analysis a Belt must use the formula shown to determine if a certain value is between -1 and + 1 which will lead to a conclusion relative to the hypothesis. The value calculated by this formula is the ____________ .
(A) Proportion of +/- 2 Standard Deviations to the total spread of the hypothesis data.
(B) Population Correlation Coefficient
(C) Sample Correlation Coefficient
Answer: (c) Sample Correlation Coefficient. This is a definition question. See article here: Correlation Coefficient.
Question: A battery manufacturer was considering changing suppliers for a particular part. The purchasing manager required that the average cost of the part be less than or equal to $32 in order to stay within budget. A sample of the 32 initial deliveries had a Mean of the new product upgrade price of $28 with an estimated Standard Deviation of $3. Based on the data provided, the Z value for the data assuming a Normal Distribution is?
A) 0.67
B) 1.33
C) 2.67
D) 4.33
Answer: 1.33 This is an easy algebra question. Z = (X – µ)/σ = 28-32 / 3 = 4/3 = 1.33. See Z Scores.
Question: When analyzing sample data a Belt may experience a Bimodal Distribution with each mode displaying Normal Distribution. This could be caused by _____________ .
A: Two different machines being read
B: Two operators on different shifts
C: Two suppliers parts being used
D: All these are correct answers
Answer: All of these answers are correct. If you see a bimodal distribution, you are most likely seeing a depiction of 2 different processes. 2 machines, supplier parts, operators, shifts could all ultimately be 2 or more distinct processes. See Bimodal.
Your Assignment
Assignment for everyone
A reminder that if you want to take these questions yourself you can via my online system. If you haven’t take then baseline exam yet, do so. You can find one available in all of my courses both paid and free.
As great as the baseline practice exams aren’t enough. You need to find more questions.
Your assignment this week is to find and practice more questions for these topics.
Assignment for PYSSGB Students
If you are a Pass Your Six Sigma Green Belt member, take all of the individual topic quizzes as well as the the cumulative exams for Module 7. Let me know what you think. (Want to be a member? Sign up here.)
What Do You Think?
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Next Time
Next week we will dive into the Improve and Control Phases of DMAIC.
I’d love feedback on how you thought the second week went. What was good, what could be better?
