Math 2200 - Chapter 9

9 Hypothesis Testing for a Single Mean

Definition: Null Hypothesis
A hypothesis to be tested (H_o).

Definition: Alternative Hypothesis
- A hypothesis considered alternate to the null hypothesis (H_a).

Note: the null hypothesis is always expressed as: H_o: m = m_o

Properties of Ha.
    If H_a: m ¹ m_o , our hypothesis is two-tailed
    If H_a: m < m_o , our hypothesis is left-tailed
    If H_a: m > m_o , our hypothesis is right-tailed

*Left- and right-hand tests are one-tailed

Section 9.2 Hypothesis Testing Terminology

Definition: Test statistic
quantity computed from sample data

Definition: Rejection region
set of values for test statistic that allow rejection of null hypothesis.

Definition: Nonrejection (acceptance) region
set of values for test statistic that do not allow rejection of null hypothesis

Definition: Critical values
separate the rejection and nonrejection region.

Note: z =

is our test statistic.

Definition: Type I error
the null hypothesis is rejected when it is true.

Definition: Type II error
the null hypothesis is accepted when it is false.

Definition: Significance level
(a ) is the probability of making a Type I error. (i.e., a true H_o is rejected)

Note:
    i. if H_o is rejected, conclude that H_a is probably true.
    ii. if H_o is not rejected, conclude the data do not provide enough evidence to support H_a.

   If H_o is rejected at significance level a, we say "the results are statistically significant at the level a ".
   If H_o is not rejected at significance level a, we say "test was not statistically significant at the level a".

Section 9.3 Large-Sample Hypothesis Tests for One Population (Assumption: n ³ 30)

Theorem: For significance level a , a is the probability that the test statistic will be in the rejection region if H_o is true

Procedure: One sample z-test for a population mean (critical value approach)
Assume:
1. normal population or large sample (n ≥ 30)
2. s is known
    i. State H_o and H_a
    ii. Decide on significance level, a
    iii. Find the critical values (±z_a/2, -z_a , z_a )
    iv. Compute the test statistic (z =

)
v. Reject or do not reject H_o
vi. State your conclusion

Note: statistical significance does not imply practical significance.

Section 9.4 Type II Errors and Power

Definition: power of a hypothesis test
is the probability of not making a Type II error; that is, the probability of rejecting a false null hypothesis.

Note: Power = 1 – P(Type II error) = 1 – β.

If the power is near 0, the hypothesis test is not very good at detecting a false null hypothesis.
If the power is near 1, the hypothesis test is extremely good at detecting a false null hypothesis.

9.5 - p - values

Definition: The p-value
is the probability of observing a test statistic as inconsistent (or more) with H_o than the test statistic actually observed (i.e., what if another sample had a mean even further away from m in the rejection region)

Note: The p-value is referred to as the "observed significance level".

Procedure: One sample z-test for a population mean (P-value approach)
Assume:
1. normal population or large sample (n ≥ 30)
2. s is known
    i. State H_o and H_a
    ii. Decide on significance level, a
    iii. Compute the test statistic ( z = )
    iv. Determine the P-value
    v. If P ≤ α, reject H_o; otherwise, do not reject H_o.
    vi. State your conclusion

Section 9.6 Hypothesis Test for Normal Population Mean
(Assumption: the population is normally distributed) *regardless of sample size use t-table

Procedure: One sample t-test for a population mean (critical value approach)
Assume:
1. normal population or large sample (n > 30)
2. s is known

    i. State H_o and H_a
    ii. Decide on significance level, a
    iii. Find the critical values (±t_a/2 , -t_a , t_a ) with df = n - 1
    iv. Compute the test statistic (t = )
    v. Reject or do not reject Ho
    vi. State your conclusion

Procedure: One sample t-test for a population mean (P-value approach)
Assume:
1. normal population or large sample (n ≥ 30)
2. s is unknown
   i. State H_o and H_a
   ii. Decide on significance level, α
   iii. Compute value of the test statistic; denote the value as t_o
   iv. Estimate the P-value
   v. If P ≤ α , reject Ho; otherwise, do not reject Ho
   vi. State your conclusion

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