Main > Critical Values & Hypothesis Testing

## When to reject the null hypothesis

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

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

General Situations

General Situations: P Value

A Proportion

A Proportion (P Value Method)

## Support or Reject Null Hypothesis for a General Situation

Deciding to support or reject null hypothesis can seem like an overwhelming task at first. The question will state that a researcher made a claim; your task is to decide whether the evidence supports the claim.

**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 Hypothesis.

**Step 2:** *Find the critical value. *

You should have already covered that topic by this point, but here is Find a critical value as a reminder.

**Step 3:*** Draw a normal distribution with your critical value.* Shade in the appropriate area (i.e. for a right-tailed test or a left-tailed test, shade in the tail).

**Step 4:** *If you already have the population standard deviation, go straight to step 6. If you do not have the population standard deviation (i.e. you are presented with a list of data) go to step 5*.

**Step 5:*** Find the mean (*)* and standard deviation(s) for the data.*

**Step 6:*** Use the following formula to find the z-score. *If you have the population standard deviation σ, use it instead of the sample standard deviation, *s.*

If formulas confuse you, all this formula is asking you to do is:

- Subtract the the null hypothesis mean (μ–you figured this out in step 1) from the mean of the data ( from step 5). Set this number aside for a moment.
- Divide the standard deviation (σ or s) by the square root of your sample size (the question either stated your sample size or you were given a certain amount n of data). For example, if thirty six children are in your sample and your standard deviation is 2, then 3/√36=0.5
- Divide your result from step 1 by your result from step 2 (i.e. step 1/step2)

**Step 7:*** *Compare your answer from step 6 with the α value given in the question. Should you support or reject null hypothesis? If step 6 is less than α, reject the null hypothesis, otherwise do not reject it.

That’s it!

## 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 null hypothesis.

**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. *

**Step 3:*** Draw a normal distribution with your critical value.* Shade in the appropriate area (i.e. for a right- or left-tailed test, shade in the tail).

**Step 4:** *If you already have the population standard deviation, go straight to step 6. If you do not have the population standard deviation (i.e. you are presented with a list of data) go to step 5*.

**Step 5:** Calculate the mean and standard deviation(s) for your data.

**Step 6:***. Use the following formula to find the z-value. *If you have the population standard deviation σ, use it instead of the sample population, *s.*

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

If formulas confuse you, all this formula is asking you to do is:

- Subtract the the null hypothesis mean (μ–you figured this out in step 1) from the mean of the data ( from step 5). Set this number aside for a moment.
- Divide the standard deviation (σ or s) by the square root of your sample size (the question either stated your sample size or you were given a certain amount n of data). For example, if thirty six children are in your sample and your standard deviation is 2, then 3/√36=0.5
- Divide your result from step 1 by your result from step 2 (i.e. step 1/step2)

*/ ?>

**Step 7:** Find the P-Value by looking up your answer from step 6 in the z-table.** **Note: for a two-tailed test, you’ll need to double this amount to get the P-Value.

**Step 8:** Compare your answer from step 7 with the α value given in the question. Should you support or reject null hypothesis? If step 7 is less than α, reject the null hypothesis, otherwise do not reject it.

That’s it!

## 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 16% of vegetarians are actually vegans. In a recent survey, 19 out of 100 vegetarians stated they were vegan. 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”)*. H_{o}:p=0.16 (claim); H_{1}:p≠0.16

**Step 2:** *Find the critical value. *

**Step 3:** *Compute * by dividing the number of positive respondents from the number in the random sample:

19/100 = 0.19.

**Step 4:***Find ‘p’ by converting the stated claim to a decimal:*

16%=0.16.

Also, find ‘q’ by subtracting from 1: 1-0.16=0.84.

**Step 5:** *Use the following formula to calculate your test value.*

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

All this is asking you to do is:

- Subtract p from(0.19-0.16=0.03). Set this number aside.
- Multiply p and q together, then divide by the number in the random sample. (0.16 x 0.84) / 200 = 0.000672
- Take the square root of your answer to 2
*. sqrt(*0.000672)=*0.0259* - Divide your answer to 1. by your answer in 3. 0.03/0.0259=1.158.

*/ ?>

**Step 6:** *Compare your answer from step 5 with the α value given in the question*. Support or reject null hypothesis? If step 5 is less than α, reject the null hypothesis, otherwise do not reject it. In this case, 1.158 is not less than our α, so we do not reject the null hypothesis.

## Support or Reject Null Hypothesis for a Proportion: P Value Method

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”)*. H_{o}:p ≤ 0.23; H_{1}: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:

- Subtract p from(0.3-0.23=0.07). Set this number aside.
- Multiply p and q together, then divide by the number in the random sample. (0.23 x 0.77) / 420 = 0.00042
- Take the square root of your answer to 2
*. sqrt(*0.1771)=*0.*0205 - 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-value 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.

Check out our Youtube channel for video tips!

Have questions on how to support or reject null hypothesis? Post your question in our forum!

I’m having a really hard time with this. I keep rereading the steps and they make sense as I try the example you have provided, but as soon as I work the homework/test questions, I’m getting the answer wrong. I just can’t figure out where I’m going wrong. So far this site has been very helpful but I’m lost with this chapter.

Jennifer,

Email me the problem you are working on and I will take a look,

Stephanie

One thing I do not understand is why Ho:p ≤ 0.23. I would think it would be greater than or equal to because the question states “A researcher claims that more than”. Great. I am confused on the first step…

Angie,

That’s because the null hypothesis (Ho:p ≤ 0.23)is the opposite of the claim.

Stephanie

I have to admit, this seemed a little overwhelming when first looking at all of the steps involved. But, your explanation was easy to follow and very helpful when doing the homework. Thank You.

This is confusing. Theses steps confused me more. I have been using “show me” on the assignments and trying to make sense of it.

Yeah, this stuff isn’t easy.

THIS HAS BY FAR BEEN MY MOST TRYING SECTION. AND ITS NOT EVEN THE WORK ITS THE LAST PART, THE WORDING. I GET SO LOST ON WETHER TO SUPPORT OR REJECT THE NULL. THE BOOK WORDS IT DIFF THAN THE HOMEWORK AND THE ‘HINT’ SECTION ON THE HOMEWORK IS NO HELP.

IF ANYONE CAN HELP PLZ LET ME KOW ID BE FOREVER GREATFUL!! :)

I agree that the wording is very confusing. I understand how to do the equations. But, the wording at the end of the problem is really confusing to me as well.

I agree with everyone else I don’t undestand the wording or anything about these hypothesis.

I find these hypothesis word problems very confusing. I have done multiples problems and everything matches up except for my “yes” or “no” response. Maybe I am not understanding how to answer when you reject the hypothesis. It does not make since that you would then say yes to the claim????

I believe Step 3 should = .77 instead of .67

You are correct :)

Thanks, Jennifer!

For some reason my calculations do not add up. I will just have to keep practicing.

What happened to Step 4??

yes this had to be the hardest section, but any little help is good.

I liked this one because it finally helped me to understand the P-value method, I was doing it backwards from how it should be understood in terms of Ha and Ho but now I understand.

i am confused why the The z-value for 3.41 is .4997. my table says the z value for 3.41 is .99996

You’re looking at the wrong z-table for the problem. There are two tables — one for the left of the curve and one for right of the curve.

This is very helpful for me, I finally understand how to answer such a question I the exam, but I don’t understand where the 0.500 is from

And why to substract it from the z value ?!please clear this up for me as am

Just learning about hypothesis testing, I’d also appreciate if you’d explain to me more about the z tables , are they like standard tables?! For all hypothesis testing ? Am a lil lost so please help!!:(

I think this forum post explains really well why you subtract the z-value from .5:

http://statisticshowto.com/forums/viewtopic.php?f=2&t=272&p=301#p301

It gets better with practice!

The z-tables aren’t used for all hypothesis testing (there are other tables!). If you need further help, feel free to ask on the forums as I don’t get to comments as often as I’d like :)

Stephanie

Step 5 reads

Find the P-Value by looking up your answer from step 5 in the z-table. The z-value for 3.41 is .4997. Subtract from 0.500: 0.500-.4977=0.023.

Why does it indicate subtract .4997 from .5, but it instead subtracts .4977 from .5?

I believe this is a mistake and should be corrected.

Ed,

Thanks for catching that typo!

Stephanie

I’m a bit confused too. On my Standard Normal Distribution table, it shows the z-value for 3.41 to be .9997. Where do you get .4997 or .4977- either way I’m not seeing it.

Kris,

There are a couple of different versions of the z-table. The sample problems on this site use the z-table that is right of z . *Some* stats books have the whole table…that’s probably what you are looking at. In any case, you would still get the same answer (1-.9999=.03). You just subtract from 1, not .5 if you are using a “whole z” table.

Stephanie

thanks

thank you

how to differentiate different types of z-tables?

Hi, Ize,

Thank you for your question. Unfortunately, time constraints prevent me from answering math questions in the comments. Could you post your question on our forums? One of our mods would be glad to help.

Stephanie