Chapter 10 Key points
A.P. Statistics
[10-1]
Definitions
The null hypothesis, denoted by H0, is a claim about a population characteristic that is
initially assumed to be true.
The alternative hypothesis, denoted by Ha, is the competing claim.
~In carrying out a test of H0 versus Ha, the hypothesis H0 will be rejected in favor of Ha
only if sample evidence strongly suggests that H0 is false. If the sample does not contain
such evidence, H0 will not be rejected. The two possible conclusions are then reject H0 or
fail to reject H0.
~The form of a null hypothesis is
H0: population characteristic = hypothesized value
where the hypothesized value is a specific number determined by the problem context.
~The alternative hypothesis will have one of the following three forms:
Ha: population characteristic > hypothesized value
Ha: population characteristic < hypothesized value
Ha: population characteristic ≠ hypothesized value
[10-2]
Definitions
Type I error the error of rejecting H0 when H0 is true
Type II error the error of failing to reject H0 when H0 is false
The probability of a type I error is denoted by α and is called the level of significance of
the test. Thus, a test with α = .01 is said to have a level of significance of .01 or to be a
level .01 test.
The probability of a type II error is denoted by β.
After assessing the consequences of type I and type II errors, identify the largest α that is
tolerable for the problem. Then employ a test procedure that uses this maximum
acceptable value—rather than anything smaller—as the level of significance (because
using a smaller α increases β). In other words, don’t make α smaller than it needs to be.
[10-3]
Definitions
A test statistic is the function of sample data on which a conclusion to reject or fail to
reject H0 is based.
The P-value (also sometimes called the observed significance level) is a measure of
inconsistency between the hypothesized value for a population characteristic and the
observed sample. It is the probability, assuming that H0 is true, of obtaining a test statistic
value at least as inconsistent with H0 as what actually resulted.
~A decision as to whether H0 should be rejected results from comparing the P-value to
the chosen α:
~H0 should be rejected if P-value < α.
H0 should not be rejected if P-value > α.
~Determination of the P-value When the Test Statistic Is z
1. Upper-tailed test
Ha: π > hypothesized value
2. Lower-tailed test
Ha: π < hypothesized value
3. Two-tailed test
Ha: π ≠ hypothesized value
Summary of Large-Sample z Test for π
Null hypothesis:
H0: π = hypothesized value
Test statistic:
z
Alternative hypothesis:
Ha: π > hypothesized value
Ha: π < hypothesized value
Ha: π ≠ hypothesized value
p  hypothesized value
(hypothesized value)(1 - hypothesized value)
n
P-value:
Area under z curve to right of calculated z
Area under z curve to left of calculated z
(i) 2(area to right of z) if z is positive
(ii) 2(area to left of z) if z is negative
Assumptions:
1. p is the sample proportion from a random sample.
2. The sample size is large. This test can be used if n satisfies both
n(hypothesized value) > 10 and n(1 - hypothesized value) > 10.
3. If sampling is without replacement, the sample size is no more than
10% of the population size.
Steps in a Hypothesis-Testing Analysis
1.
Describe the population characteristic about which hypotheses are to be tested.
2.
State the null hypothesis, H0.
3.
State the alternative hypothesis, Ha.
4.
Select the significance level α for the test.
5.
Display the test statistic to be used, with substitution of the hypothesized value identified in step 2
but without any computation at this point.
6.
Check to make sure that any assumptions required for the test are reasonable.
7.
Compute all quantities appearing in the test statistic and then the value of the test statistic itself.
8.
Determine the P-value associated with the observed value of the test statistic.
9.
State the conclusion (which will be to reject H0 if P-value < α and not to reject H0 otherwise). The
conclusion should then be stated in the context of the problem, and the level of significance should
be included.
Steps 1– 4 constitute a statement of the problem, steps 5–8 give the analysis that will lead to a decision, and
step 9 provides the conclusion.
[10-4]
The One-Sample t Test for a Population Mean
Null hypothesis:
Test statistic: t 
H0: μ = hypothesized value
x  hypothesized value
s
n
Alternative hypothesis:
Ha: μ > hypothesized value
Ha: μ < hypothesized value
Ha: μ ≠ hypothesized value
P-value:
Area to right of calculated t under t curve with df = n – 1
Area to the left of calculated t under t curve with df = n - 1
(i) 2(area to right of t) if t is positive
(ii) 2(area to left of t) if t is negative
Assumptions: 1. x and s are the sample mean and sample standard deviation from a
random sample.
2. The sample size is large (generally n > 30) or the population
distribution is at least approximately normal.
[10-5]
The power of a test is the probability of rejecting the null hypothesis.
Effect of Various Factors on the Power of a Test
1. The larger the size of the discrepancy between the hypothesized value and the true
value of the population characteristic, the higher the power.
2. The larger the significance level, a, the higher the power of the test.
3.
The larger the sample size, the higher the power of the test.
When H0 is false, power = 1 - β.
When the population distribution is normal, the t test for testing hypotheses about μ has
smaller β than does any other test procedure that has the same level of significance α.