Chapter 7 Notes (Condensed)
Hypotheses
Null Hypothesis (H0) – ALWAYS contains an element of equality (Either ≥, ≤, or =)
Alternate Hypothesis (Ha) – ALWAYS strictly inequality (Either <, >, ≠)
The alternate hypothesis is the COMPLEMENT of the null hypothesis.
If H0 is ≥, then Ha is <.
If Ha is <, it is called a “left-tail test”
If H0 is ≤, then Ha is >.
If Ha is >, it is called a “right-tail test”
If H0 is =, then Ha is ≠.
If Ha is ≠, it is called a “two-tail test”
Either one of these hypotheses may be the claim – you have to read the question/statement to determine
which one represents the claim.
We ALWAYS test the NULL hypothesis (Reject or Fail to Reject)
Types of Errors
Type I – Reject the null hypothesis when it was actually true (False negative)
Type II – Fail to reject the null hypothesis when it is false (False positive)
Level of Significance (α)
The maximum allowable probability of making a Type I error. This will be given to you.
P-value test (n ≥ 30)
Find the standardized test statistic (z-score that corresponds to the sample data)
𝑥̅ −𝜇
𝑧 = ⁄ ; when n ≥ 30, you can use s for σ.
𝜎 √𝑛
Find the probability p of that z-score occurring.
Probability depends upon which type of test you are doing.
Left-tail test – find the area to the left of z.
2nd VARS normalcdf(-1E99, z)
Right-tail test – find the area to the right of z.
2nd VARS normalcdf(z, 1E99)
Two-tail test – find the area to the left of –z and to the right of z. 1 - 2nd VARS normalcdf(-z, z)
Compare p to α
If p ≤ α, you will REJECT the null hypothesis.
If p > α, you will FAIL TO REJECT the null hypothesis.
Rejection Region test (n ≥ 30)
Find the z-score that corresponds to α.
Left tail test –
zc = 2nd VARS invNorm(α)
The Rejection region is to the left of this z-score.
nd
Right tail test - zc = 2 VARS invNorm(1-α)
The Rejection region is to the right of this z-score.
Two tail test zc = 2nd VARS invNorm(α/2)
The Rejection region is outside of the ± of this z-score.
Compare the standardized test statistic to the rejection region.
If z is in the rejection region, reject the null hypothesis.
If z is not in the rejection region, fail to reject the null hypothesis.
t-test (n < 30)
Find the rejection region by using the t-distribution chart.
Degrees of freedom = n-1.
Use the column that relates to whether you have one (left or right) tail or two tail test, and α.
If you are doing a left-tail test, use the negative of the number on the chart.
Rejection region is to the left of this t-score.
If you are doing a right-tail test, use the positive of the number on the chart.
Rejection region is to the right of this t-score.
If you are doing a two-tail test, use both the positive and negative of the number on the chart.
Rejection region is to the outside of these t-scores.
To the left of –t, or to the right of t.
Find the standardized test statistic by using the same formula that you used for the z-score.
𝑥̅ −𝜇
𝑡= ⁄
𝑠 √𝑛
Use the same rules to decide whether to reject or fail to reject as you did for the z-test.
If standardized test statistic is in rejection region, reject. Otherwise, fail to reject.
Hypothesis Test for Proportions
Make sure that np and nq are both ≥ 5.
Test statistic is 𝑝̂ (the sample proportion), and the standardized test statistic is z.
𝑝̂−𝑝
𝑧=
√𝑝𝑞/𝑛
Once you have the z-score, the decision making process is the same as we’ve been doing all chapter.
Testing Guidelines for Different Types of Hypothesis Tests
P-values for a z-test for μ
(Large Sample)
Identify H0 and Ha
Identify α
Determine the
standardized test
statistic
𝑧=
𝑥̅ − 𝜇
𝜎⁄√𝑛
Rejection Regions for a ztest for μ (Large Sample)
Identify H0 and Ha
Identify α
Determine the
standardized test
statistic
𝑧=
𝑥̅ − 𝜇
𝜎⁄√𝑛
Use the z-score to find p
Use normalcdf functions
described above
Find the critical value(s)
and rejection region(s)
Use invNorm functions
described above.
If p ≤ α, Reject H0
If p > α, Fail to reject H0
If standardized test
statistic is in the
rejection region, reject
H0.
If standardized test
statistic is not in the
rejection region, fail to
reject H0
t-test for a z-test for μ
(Small Sample)
Identify H0 and Ha
Identify α
Determine the
standardized test
statistic
𝑡=
𝑥̅ − 𝜇
𝑠⁄√𝑛
Find the critical value
Use the t-distribution
chart, with degrees of
freedom of n-1, and
column relating to α and
one or two tail test.
Follow rules stated
above for whether to use
positive, negative, or
both. Also follow rules
listed above to establish
rejection region(s).
If standardized test
statistic is in the
rejection region, reject
H0.
If standardized test
statistic is not in the
rejection region, fail to
reject H0
z-test for a proportion p
(Large Sample)
Identify H0 and Ha
Identify α
Determine the
standardized test
statistic
𝑧=
𝑝̂ − 𝑝
√𝑝𝑞/𝑛
Find the critical value(s)
and rejection region(s)
Use invNorm functions
described above.
If standardized test
statistic is in the
rejection region, reject
H0.
If standardized test
statistic is not in the
rejection region, fail to
reject H0