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College Prep Stats
Chapter 8 Important Info
Section 8.2
Null Hypothesis: H0
The null hypothesis (denoted by H0) is a statement that the value of a population parameter (such as proportion or mean) is equal to
some claimed value.
H0: p = some decimal
OR
H0: µ = some number
Alternative Hypothesis: H1
The alternative hypothesis (denoted by H1 or Ha or HA) is the statement that the parameter has a value that somehow differs from the
null hypothesis.
The symbolic form of the alternative hypothesis must use one of these symbols: , <, >.
If the problem says:
At least, No less than,
Greater than or equal to
Use the opposite!: <
H1: p < some decimal
H1: p > some decimal
H1: p  some decimal
OR
OR
OR
H1: µ < some number
H1: µ > some number
H1: µ  some number
If the problem says:
At most, No more than,
Less than or equal to
Use the opposite!: >
Critical Region
The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis.
*If the test were left tailed, then any test
statistic value below the critical value should
cause you to reject H0.
*If the test were two tailed, then any test
statistic value below the left critical value or
above the right critical value should cause you
to reject H0.
*
Rejection region
Any test statistic value above
the critical value should cause
you to reject H0.
Significance Level
The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is
actually true.
Critical Value
Two-tailed Test
When H0 uses “=” and H1: “” (means less than or greater than)
α is divided equally between the two tails of the critical region
z* = ±critical value … (invNorm(area in the left tail, 0, 1))
-z*
Left-tailed Test
When H0 uses “=” and H1 uses “< ” (if you turn the “less than” symbol into an arrow, it points LEFT)
α is the left tail
z*
z* = –critical value… (invNorm(area in the left tail, 0, 1))
Right-tailed Test
When H0 uses “=” and H1 uses “> “ (if you turn the “greater than” symbol into an arrow, it points RIGHT)
α is the right tail
z* = +critical value (invNorm(1 – area in the right tail, 0, 1))
Test Statistic
Test Statistic for proportion: z 
p̂  p
pq
n
where
𝑥
𝑝̂ = sample proportion =
𝑛
p = population proportion
q=1–p
n = sample size
P-Value
The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the
one representing the sample data, assuming that the null hypothesis is true.
Critical region in the left tail:
P-value = area to the left of the test statistic
Critical region in the right tail:
P-value = area to the right of the test statistic
Critical region in two tails:
P-value = twice the area in the tail beyond the test statistic
Remember to find probabilities use the normalcdf feature in your calculator:
For right-tailed tests: P(z > test statistic)
*To find this probability in your calculator, type: normalcdf(z test statistic, 99999, 0, 1)
For left-tailed tests: P(z < –test statistic)
*To find this probability in your calculator, type: normalcdf(–99999, –z test statistic, 0, 1)
***Don’t forget if your test is two-sided, double your P-value.***
Conclusions in Hypothesis Testing
Decision Criterion
P-value method:
Using the significance level, :
Traditional method:
Use your critical value to create a critical region.
If P-value   , reject H0.
If P-value >  , fail to reject H0.
If the test statistic falls within the critical region, reject H0.
If the test statistic does not fall within the critical region, fail to reject H0.
Wording of Final Conclusion
There is enough
evidence to suggest that
the … (alternative
hypothesis in context)
There is not enough
evidence to suggest that
the … (alternative
hypothesis in context)
Section 8.3
Test the claim about a proportion.
Write a null and alternative hypothesis.
Find the critical value, test statistic, and p – value, using the information from Section 8.2.
Make a conclusion. Will you reject H0 or fail to reject H0?
Section 8.4
Test the claim about a mean, with  known.
Write a null and alternative hypothesis.
Find the critical value and p – value using the information from Section 8.2.
Find the test statistic using the formula below.
Make a conclusion. Will you reject H0 or fail to reject H0?
Notation
n = sample size
x  sample mean
x  population mean of all sample means from samples of size n
 = known value of the population standard deviation
Test Statistic for Testing a Claim about a Mean (with  Known)
z
x 

n
Section 8.5
Test the claim about a mean, with  unknown.
Write a null and alternative hypothesis.
Find the critical value, test statistic, and p – value, using the information below.
Make a conclusion. Will you reject H0 or fail to reject H0?
Notation
n = sample size
x  sample mean
x  population mean of all sample means from samples of size n
s = sample standard deviation
df = degrees of freedom (n – 1)
Critical Value
Two-tailed Test
When H0 uses “=” and H1: “” (means less than or greater than)
α is divided equally between the two tails of the critical region
t* = ±critical value …(invT(area in the left tail, df))
-t*
Left-tailed Test
When H0 uses “=” and H1 uses “< ” (if you turn the “less than” symbol into an arrow, it points LEFT)
α is the left tail
t* = –critical value… (invT(area in the left tail, df))
t*
Right-tailed Test
When H0 uses “=” and H1 uses “> “ (if you turn the “greater than” symbol into an arrow, it points RIGHT)
α is the right tail
t* = +critical value (invT(1 – area in the right tail, df))
Test Statistic for Testing a Claim about a Mean (with  Not Known)
t
x 
s
n
P-values:
Use the tcdf feature on your calculator, with degrees of freedom (df) = n – 1.
For right tailed test:
*To find this probability in your calculator, type: tcdf(t test statistic, 99999, df)
For left tailed test:
*To find this probability in your calculator, type: tcdf(–99999, –t test statistic, df).
***Don’t forget, if your test is two-sided, double your P – value!!!!***
z* and z VERSUS t* and t
Choosing between the Normal and Student t Distributions when Testing a Claim about a Population Mean µ
Use the Student t distribution when  is not known and either or both of these conditions is satisfied:
The population is normally distributed or n > 30.