# Stats without Tears

Seven Steps of Hypothesis Tests

Updated 3 Nov 2020
(What’s New?)

Copyright © 2010–2023 by Stan Brown, BrownMath.com

Seven Steps of Hypothesis Tests

Updated 3 Nov 2020
(What’s New?)

Copyright © 2010–2023 by Stan Brown, BrownMath.com

Print:

**Advice**: Always number your steps. That helps others
find the key features of your test, and you don’t forget any
steps.

**See also:**
Inferential Statistics: Basic Cases

Top 10 Mistakes of Hypothesis Tests

Following are patterns for your hypotheses in the cases covered in the text. With Cases 1 through 5, if you can say anything meaningful about the consequences if each hypothesis is true, add that.

Bad example (adds little or nothing to the symbols):

H_{0}: μ = 67.6, average 2-liter bottle contains 67.6 fl oz

H_{1}: μ < 67.6, average 2-liter bottle contains less than 67.6 fl oz

Good example (explains the implications):

H_{0}: μ = 67.6, average bottle filled properly

H_{1}: μ < 67.6, average bottle is underfilled

In Cases 1 through 5, a test for < or > is called a one-tailed test, and a test for ≠ is called a two-tailed test. Please see One-Tailed or Two-Tailed? for advice on choosing between them.

**Case 1:**
(Testing mean of one population against a number
called μ_{o})

H_{0}: μ = *number*

H_{1}: μ < *number* *or* μ ≠ *number* *or* μ > *number*

**Case 2:**
(Testing proportion in one population
against a number called *p*_{o})

H_{0}: *p* = *number*

H_{1}: *p* < *number* *or* *p* ≠ *number* *or* *p* > *number*

**Case 3:**
(Testing mean difference (paired data))

d = _____ − _____

H_{0}: μ_{d} = 0

H_{1}: μ_{d} < 0 *or* μ_{d} ≠ 0 *or* μ_{d} > 0

**Case 4:**
(Testing difference of independent means)

pop. 1 = _____, pop. 2 = _____

H_{0}: μ_{1} = μ_{2}

H_{1}: μ_{1} < μ_{2} *or* μ_{1} ≠ μ_{2} *or* μ_{1} > μ_{2}

**Case 5:**
(Testing difference of population proportions)

pop. 1 = _____, pop. 2 = _____

H_{0}: *p*_{1} = *p*_{2}

H_{1}: *p*_{1} < *p*_{2} *or* *p*_{1} ≠ *p*_{2} *or* *p*_{1} > *p*_{2}

**Case 6:**
(Testing goodness of fit)

H_{0}: The _____ model is consistent with the data.

H_{1}: The model is not consistent with the data.

**Case 7:**
(Testing independence)

H_{0}: _____ and _____ are independent.

H_{1}: _____ and _____ are dependent.

**Case 7:**
(Testing homogeneity)

H_{0}: The proportions are all equal.

H_{1}: Some proportions are different from others.

Short and sweet:

α = _____

Please see Inferential Statistics: Basic Cases for specific requirements. For Cases 6 and 7, it’s easier to check requirements if you move this step after Steps 3/4.

Show screen name. Example: `T-Test`

. You
don’t need to write down keystrokes, such as “STAT TESTS
2”.

Show all inputs.

Show *new* outputs, meaning any that weren’t on
the input screen.

No room for creativity here. Write down whichever one of these applies:

p < α. Reject H_{0} and accept H_{1}.

p > α. Fail to reject H_{0}.

Here you have a lot of latitude as long as you state the correct conclusion in English and give the significance level or p-value, or both.

If you rejected H_{0}, state H_{1} without doubting words
like *may* or *could*. Examples:

At the 0.05 significance level, the average 2-liter bottle contains less than 67.6 fl oz. Drinkems is underfilling the bottles.

Or,

The average 2-liter bottle
contains less than 67.6 fl oz. Drinkems is underfilling
the bottles (*p* = 0.0246).

If you failed to reject H_{0}, state your non-conclusion in
neutral language, using phrases like *can’t determine
whether* or *it’s impossible to say whether*.
Examples:

At the 0.05 significance level, we can’t tell whether Drinkems is underfilling the bottles or not.

Or,

We can’t tell whether Drinkems is underfilling
the bottles or not (*p* = 0.1045).

**3 Nov 2021**: In the last example in step 6, changed p-value in the fail-to-reject outcome, so that failing to reject makes sense against the 0.05 significance level.- (intervening changes suppressed)
**30 June 2013**: New article, formed by just the reference material from the old “Hypothesis Tests: Six Steps (Plus One)”, which was written in 2010.

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