Fail To Reject Null Hypothesis Statement Example

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Subcultures have languages all their own. Teen gangs, statisticians, gamers, music buffs, sports nuts, furries...all use terminology that baffles outsiders.The arcane language helps identify kindred spirits: using the correct phrase proves you belong. The proper buzzwords can gain you admittance to the right professional circles...or the wrong biker bars. Maybe both. 

Not knowing them can get you into serious trouble. When you enter a dangerous place (like the data analysis arena), you need at least a basic grasp of the jargon the local toughs use. 

I'm not comparing any particular group of statisticians to a street gang, but the discipline definitely has its own language, one that can seem inpenetrable and obtuse. It's all too easy for a seasoned vet of the stats battlefield to confound newcomers who aren't hep to the lingo of data analysis. 

Like that gent over there...the big guy wearing the Nulls Angels jacket, the analyst everyone calls "Tiny." He's always telling war stories about how he "failed to reject the null hypothesis." 

Looking at the phrase from a purely editorial vantage, "failing to reject the null hypothesis" is cringe-worthy. Doesn't "failure to reject" amount to a double negative? Isn't it just a more high-falutin', circular equivalent to accept? At minimum, "failure to reject" is clunky phrasing.  

Maybe so. But from a statistical perspective, it's undeniably accurate. Replacing "failure to reject" with "accept" would be wrong. 

In this case, Tiny and the rest of those bad-boy statisticians in the Nulls Angels have a good reason to talk the way they do. 

What Is the Null Hypothesis, Anyway? 

There are many different kinds of hypothesis tests, including one- and two-sample t-tests, tests for association, tests for normality, and many more. If you're using Minitab statistical software, you have direct access to all of these tests through the Stat menu. If you want a little statistical guidance, the Assistant can lead you through many of the most commonly used hypothesis tests step-by-step.

In a hypothesis test, you're going to look at two propositions: the null hypothesis (or H0 for short), and the alternative (H1). The alternative hypothesis is what we hope to support. The null hypothesis, in contrast, is presumed to be true, until the data provide sufficient evidence that it is not. 

A similar idea underlies the U.S. criminal justice system: you've heard the phrase "Innocent until proven guilty"? In the statistical world, the null hypothesis is taken for granted until the alternative is proven true. The null hypothesis is never proven true; you simply fail to reject it.

How Do We "Fail to Reject" the Null Hypothesis? 

The degree of statistical evidence we need in order to “prove” the alternative hypothesis is the confidence level. The confidence level is simply 1 minus the Type I error rate (alpha, also referred to as the significance level), which occurs when you incorrectly reject the null hypothesis. The typical alpha value of 0.05 corresponds to a 95% confidence level: we're accepting a 5% chance of rejecting the null even if it is true. (When hypothesis-testing life-or-death matters, we can lower the risk of a Type I error to 1% or less.)

Regardless of the alpha level we choose, any hypothesis test has only two possible outcomes: 

  1. Reject the null hypothesis (p-value <= alpha) and conclude that the alternative hypothesis is true at the 95% confidence level (or whatever level you've selected).
     
  2. Fail to reject the null hypothesis (p-value > alpha) and conclude that not enough evidence is available to suggest the null is false at the 95% confidence level.

In the results of a hypothesis test, we typically use the p-value to decide if the data support the null hypothesis or not. If the p-value is very low (typically below 0.05), statisticians say "the null must go." 

If We Don't Accept the Alternative Hypothesis, Don't We Have to Accept the Null Hypothesis? 

This still doesn't explain why a statistician can't say "we accept the null hypothesis," as a certain unnamed, wet-behind-the-ears, statistically-challenged editor might have suggested to Tiny.

Once. 

Here's the bottom line: even if we fail to reject the null hypothesis, it does not mean the null hypothesis is true. That's because a hypothesis test does not determine which hypothesis is true, or even which is most likely: it only assesses whether available evidence exists to reject the null hypothesis. 

"Null Until Proven Alternative"

Look at it in terms of "innocent until proven guilty" in a courtroom: As the person analyzing data, you are the judge. The hypothesis test is the trial, and the null hypothesis is the defendant. The alternative hypothesis is like the prosecution, which needs to make its case beyond a reasonable doubt (say, with 95% certainty).

If the evidence presented doesn't prove the defendant is guilty beyond a reasonable doubt, you still have not proved that the defendant is innocent. But based on the evidence, you can't reject that possibility

So how would that verdict be announced? It enters the court record as "Not guilty."  

That phrase is perfect: "Not guilty" doesn't mean the defendant is innocent, because that has not been proven. It just means the prosecution couldn't prove its case to the necessary, "beyond a reasonable doubt" standard. It failed to convince the judge to abandon the assumption of innocence.

If you follow that rationale, then you can see that "failure to reject the null" is just the statistical equivalent of "not guilty." In a trial, the burden of proof falls to the prosecution. When analyzing data, the entire burden of proof falls to the sample data you've collected. Just as "not guilty" is not the same thing as "innocent," neither is "failing to reject" the same as "accepting" the null hypothesis. 

So the next time you're looking to hang around at the local Nulls Angels clubhouse, remember that "failing to reject the null" is not "accepting the null." Knowing the difference just might get Tiny to buy you a drink.  

Hypothesis TestingStatistics Help

Hypothesis Testing > Support or Reject Null Hypothesis

Contents:

  1. What does it mean to reject the null hypothesis?
  2. Support or Reject the null hypothesis: Steps

What does it mean to reject the null hypothesis?

Watch the video or read the article below:

In many statistical tests, you’ll want to either reject or support the null hypothesis. For elementary statistics students, the term can be a tricky term to grasp, partly because the name “null hypothesis” doesn’t make it clear about what the null hypothesis actually is!

Overview

The null hypothesis can be thought of as a nullifiable hypothesis. That means you can nullify it, or reject it. What happens if you reject the null hypothesis? It gets replaced with the alternate hypothesis, which is what you think might actually be true about a situation. For example, let’s say you think that a certain drug might be responsible for a spate of recent heart attacks. The drug company thinks the drug is safe. The null hypothesis is always the accepted hypothesis; in this example, the drug is on the market, people are using it, and it’s generally accepted to be safe. Therefore, the null hypothesis is that the drug is safe. The alternate hypothesis — the one you want to replace the null hypothesis, is that the drug isn’t safe. Rejecting the null hypothesis in this case means that you will have to prove that the drug is not safe.


Vioxx was pulled from the market after it was linked to heart problems.

To reject the null hypothesis, perform the following steps:

Step 1: State the null hypothesis. When you state the null hypothesis, you also have to state the alternate hypothesis. Sometimes it is easier to state the alternate hypothesis first, because that’s the researcher’s thoughts about the experiment. How to state the null hypothesis (opens in a new window).

Step 2:Support or reject the null hypothesis. Several methods exist, depending on what kind of sample data you have. For example, you can use the P-value method. For a rundown on all methods, see: Support or reject the null hypothesis.

If you are able to reject the null hypothesis in Step 2, you can replace it with the alternate hypothesis.

That’s it!

When to Reject the Null hypothesis

Basically, you reject the null hypothesis when your test value falls into the rejection region. There are four main ways you’ll compute test values and either support or reject your null hypothesis. Which method you choose depends mainly on if you have a proportion or a p-value.

Support or Reject the Null Hypothesis: Steps

Click the link the skip to the situation you need to support or reject null hypothesis for:

General Situations: P Value
P Value Guidelines
A Proportion
A Proportion (second example)

Support or Reject Null Hypothesis with a P Value


If you have a P-value, or are asked to find a p-value, follow these instructions to support or reject the null hypothesis. This method works if you are given an alpha leveland if you are not given an alpha level. If you are given a confidence level, just subtract from 1 to get the alpha level. See: How to calculate an alpha level.

Step 1:State the null hypothesis and the alternate hypothesis (“the claim”).
If you aren’t sure how to do this, follow this link for How To State the Null and Alternate Hypothesis.

Step 2:Find the critical value. We’re dealing with a normally distributed population, so the critical value is a z-score.
Use the following formula to find the z-score.

Click here if you want easy, step-by-step instructions for solving this formula.

Step 4: Find the P-Value by looking up your answer from step 3 in the z-table. To get the p-value, subtract the area from 1. For example, if your area is .990 then your p-value is 1-.9950 = 0.005. Note: for a two-tailed test, you’ll need to halve this amount to get the p-value in one tail.

Step 5: Compare your answer from step 4 with the α value given in the question. Should you support or reject the null hypothesis?
If step 7 is less than or equal to α, reject the null hypothesis, otherwise do not reject it.

P-Value Guidelines

Use these general guidelines to decide if you should reject or keep the null:

If p value > .10 → “not significant”
If p value ≤ .10 → “marginally significant”
If p value ≤ .05 → “significant”
If p value ≤ .01 → “highly significant.”

That’s it!

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Support or Reject Null Hypothesis for a Proportion

Sometimes, you’ll be given a proportion of the population or a percentage and asked to support or reject null hypothesis. In this case you can’t compute a test value by calculating a z-score (you need actual numbers for that), so we use a slightly different technique.

Sample question: A researcher claims that Democrats will win the next election. 4300 voters were polled; 2200 said they would vote Democrat. Decide if you should support or reject null hypothesis. Is there enough evidence at α=0.05 to support this claim?

Step 1:State the null hypothesis and the alternate hypothesis (“the claim”).
Ho:p ≤ 0.5
H1:p > .5

Step 2:Compute by dividing the number of positive respondents from the number in the random sample:
2200/4300 = 0.512.

Step 3:Use the following formula to calculate your test value.

Where:
Phat is calculated in Step 2
P the null hypothesisp value (.05)
Q is 1 – p

The z-score is:
.512 – .5 / √(.5(.5) / 4300)) = 1.57

Step 4: Look up Step 3 in the z-table to get .9418.

Step 5: Calculate your p-value by subtracting Step 4 from 1.
1-.9418 = .0582

Step 6:Compare your answer from step 5 with the α value given in the question. Support or reject the null hypothesis? If step 5 is less than α, reject the null hypothesis, otherwise do not reject it. In this case, .582 (5.82%) is not less than our α, so we do not reject the null hypothesis.

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Support or Reject Null Hypothesis for a Proportion: Second example

Sample question: A researcher claims that more than 23% of community members go to church regularly. In a recent survey, 126 out of 420 people stated they went to church regularly. Is there enough evidence at α = 0.05 to support this claim? Use the P-Value method to support or reject null hypothesis.

Step 1:State the null hypothesis and the alternate hypothesis (“the claim”). Ho:p ≤ 0.23; H1:p > 0.23 (claim)

Step 2:Compute by dividing the number of positive respondents from the number in the random sample:
63 / 210 = 0.3.

Step 3: Find ‘p’ by converting the stated claim to a decimal:
23% = 0.23.
Also, find ‘q’ by subtracting ‘p’ from 1: 1 – 0.23 = 0.77.

Step 4: Use the following formula to calculate your test value.

Click here if you want easy, step-by-step instructions for solving this formula.

If formulas confuse you, this is asking you to:

  1. Subtract p from(0.3 – 0.23 = 0.07). Set this number aside.
  2. Multiply p and q together, then divide by the number in the random sample. (0.23 x 0.77) / 420 = 0.00042
  3. Take the square root of your answer to 2. √(0.1771) = 0.0205
  4. Divide your answer to 1. by your answer in 3. 0.07 / 0.0205 = 3.41

Step 5: Find the P-Value by looking up your answer from step 5 in the z-table.The z-score for 3.41 is .4997. Subtract from 0.500: 0.500-.4997 = 0.003.

Step 6:Compare your P-value to α. Support or reject null hypothesis? If the P-value is less, reject the null hypothesis. If the P-value is more, keep the null hypothesis.
0.003 < 0.05, so we have enough evidence to reject the null hypothesis and accept the claim.

Note: In Step 5, I’m using the z-table on this site to solve this problem. Most textbooks have the right of z-table. If you’re seeing .9997 as an answer in your textbook table, then your textbook has a “whole z” table, in which case don’t subtract from .5, subtract from 1. 1-.9997 = 0.003.

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