Why We Don’t “Accept” the Null Hypothesis
by Keith M. Bower, M.S. and James A. Colton, M.S.
Reprinted with permission from the American Society for Quality When performing statistical hypothesis tests such as a one-sample t-test or the AndersonDarling test for normality, an investigator will either reject or fail to reject the null hypothesis, based upon sampled data. Frequently, results in Six Sigma projects contain the verbiage “accept the null hypothesis,” which implies that the null hypothesis has been proven true. This article discusses why such a practice is incorrect, and why this issue is more than a matter of semantics.

Overview of Hypothesis Testing
In a statistical hypothesis test, two hypotheses are evaluated: the null (H0) and the alternative (H1). The null hypothesis is assumed true until proven otherwise. If the weight of evidence leads us to believe that the null hypothesis is highly unlikely (based upon probability theory), then we have a statistical basis upon which we may reject the null hypothesis.

A common misconception is that statistical hypothesis tests are designed to select the more likely of two hypotheses. Rather, a test will stay with the null hypothesis until enough evidence (data) appears to support the alternative.

The amount of evidence required to “prove” the alternative may be stated in terms of a confidence level (denoted X%). The confidence level is often specified before a test is conducted as part of a sample size calculation. We view the confidence level as equaling one minus the Type I error rate (α). A Type I error is committed when the null hypothesis is incorrectly rejected. An α value of 0.05 is typically used, corresponding to 95% confidence levels.

The p-value is used to determine if enough evidence exists to reject the null hypothesis in favor of the alternative. The p-value is the probability of incorrectly rejecting the null hypothesis.
The two possible conclusions, after assessing the data, are to: 1....

...because of this random factor, sample may not be exactly representative
* sampling error
* the difference between the sample mean and the population mean
* ensure that you have enough participants so that you get an accurate reflection of the population that you are interested in
* population mean (parameter), sample mean (statistic)
* the larger the samples, the closer to the population parameter the statistics will be
* probabilities
* the number possible outcomes that you are interested in divided by the total number of possible outcomes associated with an event
* nullhypothesis significance testing (NHST)
* the nullhypothesis states that there is no effect in the population of interest
* if the probability of obtaining the data is high, the nullhypothesis is true
* no effect in the population
* distribution
* normal distribution
* skewness
* the peak of the distribution is shifted away from the middle of the graph to either left or the right
* bimodal distribution – two identifiable peaks
* often the participants drawn from two populations; you would try to identify the two different populations and analyze the data from each group separately
* parametric tests
* making assumptions about the parameters of the...

...Hypothesis Testing
* Used to prove which are the factors that are actually impacting the mean or standard deviation of the project y.
* To see the impact of improvements after they are implemented
* P- value is critical in making decisions.
* To determine if the statistical hypothesis is true or false, the entire population should be examined, which becomes impossible for large sizes, Random sampling is done.
* The conclusion for the population is based upon statistical significance determined from sample data.
What is a Hypothesis?
* Is used to describe the assumption.
* Is based on the population parameters.
* Must be clearly stated for correct decision-making.
* Is proved based on that evidence from Statistical test.
NullHypothesis. Ho is a statement (assumption) about population(s) parameters.
* It is the one assumed to be true unless stated otherwise
* Generally describes the present status
Alternate Hypothesis, Ha, is the negation or compliment of the nullhypothesis.
* Generally describes a difference
Hypothesis Testing
* Let us illustrate the concept using a justice system.
* H0: Person is not guilty
* H1: Person is guilty
* Strong evidence is required to prove a person guilty, that is, to reject the Ho.
Hypothesis Testing: Types of error
* Type...

...dispute between the holier-than-thou Médecins Sans Frontières (MSF) and my conniving brother's Au & Associates FauxPharamaceutical (AFP).
3. (2 points) Given the above information, what kind of hypothesis test will you conduct? The y-test, z-test, t-test, χ2-test, F-test, G-test, or even the y-not-test? Please explain.
4. (2 points) What will be the nullhypothesis, the alternative hypothesis, and, hence, the "tailedness" of the test (left-tailed, right-tailed, or two-tailed)?
5. (10 points) What is be the corresponding test statistics?
6. (8 points) What is the corresponding p-value of the hypothesis test?
7. (12 points) What kind of conclusion can you draw from the hypothesis test you have just performed? Of course, representatives of AFP would like to have the conclusion skewed to their advantage. And so would the representatives from MSF. What would you do if you are representing AFP? But, if you are representing MSF, how would you present your argument? (Hint: Consider your argument based on significance levels.)
8. (8 points) But, wait. What if MSF actually does not know the population standard deviation in this case, would you conduct your hypothesis test differently? Just in case that you are going to perform the hypothesis differently, what would you do instead?
The following information is for...

...Take Home Test 2
1. A. NullHypothesis: There are no relations or associations among the groups’ mean scores.
Alternate Hypothesis: There is a relation or association among the student’s grade point averages and “if they rather prefer to stay at home than go out with friends”.
Correlations |
| Grade Point Average | I would rather stay at home and read than go out with my friends |
Grade Point Average | Pearson Correlation | 1 | .233 |
| Sig. (2-tailed) | | .120 |
| Sum of Squares and Cross-products | 12.667 | 5.002 |
| Covariance | .281 | .111 |
| N | 46 | 46 |
I would rather stay at home and read than go out with my friends | Pearson Correlation | .233 | 1 |
| Sig. (2-tailed) | .120 | |
| Sum of Squares and Cross-products | 5.002 | 36.457 |
| Covariance | .111 | .810 |
| N | 46 | 46 |
Based on the results of our Correlate Bivariate we see that the significance value is more than the p-value of .05 which means that the groups have no relationship between them. The significance value is .120. This means that we are going to accept the NullHypothesis and reject the Alternate Hypothesis. “I would rather stay at home and read than go out with my friends” has no relationship with the persons GPA.
B. NullHypothesis: There is no relation or association between people who rarely forget their appointment if they have...

...Running head: MULTIPLE SAMPLE HYPOTHESIS TESTING
Multiple Sample Hypothesis Testing
RES342: Research 11
June 14, 2010
Multiple Sample Hypothesis Testing
The purpose of this paper is to create a hypothesis based on two-sample tests. Two-sample tests compare two sample estimates with each other, whereas one-sample tests compare a sample estimate with a non-sample benchmark (Doane & Seward, 2007). The learning team has chosen to create a hypothesis testing using the wages and wage earners data set. The learning team has developed one business research question from which the team will formulate a research hypothesis. The business research question and testing simply involves creating two separate groups of the data set, and testing whether a difference in the mean of the earnings in both the older group, ages 42-64 and the younger group, ages 18-41 exists. To create a solid testing hypothesis, the team has formulated both a numerical and verbal hypothesis statement, conducted the five-step hypothesis test on the data, and presented a description of the test results by explaining how the discoveries from the hypothesis testing can be used to answer the research question.
Learning Team A believes that there are many factors that affect an individual’s wages. Some of factors that affect one’s wages are abilities, experience,...

...A NOTE ON HYPOTHESIS TESTING
|Significance Level |One-Sided Test |Two-Sided Test |
|0.10 |1.285 |1.645 |
|0.05 |1.645 |1.960 |
|0.01 |2.33 |2.575 |
Part A. Single-Sample Inference
1. Test Statistic for the Population Mean ((): Large Sample Test
H0: ( = (0
H1: ( ( (0
Test statistic:
[pic], where (0 = hypothesized value of (
Note that:
Sample mean = [pic]
Mean of the sample mean: E([pic]) = (
Standard error of [pic] = [pic]
Example 1
As part of a survey to determine the extent of required in-cabin storage capacity, a researcher needs to test the nullhypothesis that the average weight of carry-on baggage per person is μ 0 = 12 pounds, versus the alternative hypothesis that the average weight is not 12 pounds. The analyst wants to test the nullhypothesis at α = 0.05. The data collected for this study are:
n = 144; [pic]= 14.6; s = 7.8; For α = 0.05, critical values of z are ±1.96
H0: μ = 12 [two-tailed test]
H1: μ ≠ 12
[pic]
Since the test statistic falls in the upper...

...Executive summary
The following report aims to analyse and interpret the data set of 200 records regarding the CCResort. The given information includes booking identification number, income, number of people per booking, length of stay, age and overall expenditure.
From the booking ID it can be assumed that the selection of data is random, however as it is only partial information and not the population, the period of time in which the data is selected from would affect the end results of analysis.
The report is divided into two sections outlining the statistical analysis of data and hypothesis testing to observe if CCResort have met their 2 major key performance indicators (KPIs)
1 More than 40% of their customers stay for a full week (i.e. seven nights);
2 The average customer spends more than $255 per day in excess of accommodation costs.
Figures at a glance
This section of the report aims to give users a better understanding of the data through statistical data analysis of investigation categories including family income, expenditure habits, age distribution, the number of people per booking and their length of stay. These analysis are meaningful in giving users a better understanding of the customer base in relation to the key performance indicators.
1. Family income distribution
From the data collected, 62 families (31% of the sample) earn an income of more than $100,000 while 69% of the sample (138 bookings)...

...click “OK”. (See the 3 figures, below.)
6.
Your output should look like this.
7.
You should use the output information in the following manner to answer the question.
Step 0 : Check Assumptions The samples were taken randomly and independently of each other. The populations have approximately the same shapes (according to the boxplots). All sample sizes are at least 6 if k = 3 (smallest is 6). Hypotheses
Step 1 :
H0 : M1 = M2 = M3 (The median test scores are equal.) Ha : Not all of the medians are equal.
Step 2 : Step 3 : Step 4 :
Significance Level Rejection Region Reject the nullhypothesis if p-value ≤ 0.05. Test Statistic
α = 0.05
Note that the test statistic (KW = Chi-Square = 7.5023) is corrected for the existence of ties in the ranks of the data.
Step 5 : Step 6 :
Decision Since p-value = 0.0235 ≤ 0.05 = α, we reject the nullhypothesis. State conclusion in words At the α = 0.05 level of significance, there exists enough evidence to conclude that there is a difference among the three teaching methods based on the test scores.
...