Chapter 8 Hypothesis Testing
Chapter 8 Hypothesis Testing
Chapter 8 Hypothesis Testing
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<strong>Chapter</strong> 8 <strong>Hypothesis</strong> <strong>Testing</strong><br />
Stat 2601<br />
Section 8.1<br />
Introduction<br />
1. Basic Concepts<br />
1). A statistical hypothesis – a statement/claim/theory about a population parameter.<br />
The null hypothesis is a theory about a population parameter. It is denoted by H 0 ,<br />
and it is usually stated as H 0 : parameter = value.<br />
The alternative hypothesis is a theory that contradicts the null hypothesis. It is<br />
denoted by H 1 , and it is always specified as strict inequalities, such as , >, or < .<br />
Steps for Selecting the Null and Alternative Hypotheses<br />
I. Select the alternative hypothesis as that which the sampling experiment is intended<br />
to establish. The alternative hypothesis has one of the three forms:<br />
H 1 : μ > value (right-tailed)<br />
H 1 : μ < value (left-tailed)<br />
H 1 : μ value (two-sided)<br />
II. Select the null hypothesis which is usually specified as equality.<br />
Example 1: A researcher is interested in finding out whether the average age of all cars in<br />
use is higher than 8 years. The null and alternative hypotheses can be stated as:<br />
H 0 : μ = 8 and H 1 : μ > 8 . This test is called a right-tailed test.<br />
Example 2: According to a report, the mean monthly bill for cell phone users in the United<br />
States was $49.70 in 1999. We want to determine whether last year’s mean monthly bill<br />
for cell phone users has decreased from the 1999 mean of $49.70. The hypotheses are<br />
H 0 : μ = 49.70 and H 1 : μ < 49.70 . This test is called a left-tailed test.<br />
Example 3: A researcher wishes to find out whether the mean body temperature of<br />
humans is 98.6°F. The null and alternative hypotheses can be stated as:<br />
H 0 : μ = 98.6° and H 1 : μ 98.6°. This test is called a two-tailed test.<br />
2). A test statistic or test value– is a numerical value obtained from a sample and is used<br />
to decide whether the null hypothesis should be rejected.<br />
3). The critical or rejection region – the set of values for the test statistic that leads to<br />
rejection of the null hypothesis.<br />
Critical values – the values that separate the rejection from the non-rejection regions.<br />
4). Type I error – rejecting the null hypothesis when it is in fact true.<br />
Type II error – not rejecting the null hypothesis when it is in fact false.<br />
H 0 is true<br />
H 0 is false<br />
Do not reject H 0 Correct decision Type II error<br />
Reject H 0 Type I error Correct decision<br />
5). Significance level - the probability of making a Type I error, denoted by α.<br />
Three common values for significance level: 0.01, 0.05, and 0.10.<br />
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Section 8.2<br />
z Test for a Mean (Traditional Method and P-value Method)<br />
Stat 2601<br />
1. Definition<br />
The z test is a statistical test which can be used when the population standard deviation is<br />
known and the population is normally distributed or when n ≥ 30. The test statistic for a z<br />
test with null hypothesis H 0 : μ = μ 0 is:<br />
X<br />
z<br />
0 .<br />
/ n<br />
2. Traditional Method (Critical-Value Method) for <strong>Hypothesis</strong> <strong>Testing</strong><br />
1) Steps:<br />
State the hypotheses.<br />
Compute the value of the test statistic.<br />
Find the rejection region at a specified significance level α.<br />
Make the decision to reject or not reject H 0 .<br />
Interpret the results of the hypothesis test.<br />
2) Finding the Rejection Region for a Specific α value when a z-test is<br />
performed.<br />
a) Right-tailed<br />
Rejection region: z > z α , where z α is a z value that will give an area of α in the<br />
right tail of the standard normal distribution. z α is also called the critical value, a<br />
value that separates the rejection region and non-rejection region.<br />
If α = .05, the rejection region is z > 1.645.<br />
Right tail area α<br />
0<br />
C.V.<br />
b) Left-tailed<br />
For a left-tailed test, the rejection region: z < -z α .<br />
When α=.05, the C.V. is -1.645, the rejection region is z < -1.645<br />
.<br />
0<br />
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Stat 2601<br />
c) Two-sided<br />
For a two-sided test, the rejection region: z > z α/2 or z < -z α/2 .<br />
When α=.05., the rejection region is z > 1.96 or z < -1.96.<br />
Total tail area =<br />
0<br />
Table 8.1 Summary of rejection region at a specific significance level α<br />
Rejection Region<br />
Upper-tailed Lower-tailed Two-sided<br />
α = .01 z > 2.33 z < -2.33 z > 2.575 or z < -2.575<br />
α = .05 z > 1.645 z < -1.645 z > 1.96 or z < -1.96<br />
α = .1 z > 1.28 z < -1.28 z > 1.645 or z < -1.645<br />
Example 1: A researcher is interested in finding out whether the average age of all<br />
cars in use is higher than 8 years. A random sample of 40 cars were selected and the<br />
average age of cars was found to be 8.5 yrs, and the standard deviation was 3.5 yrs. At<br />
α=.05, can it be concluded that the average age of all cars in use is higher than 8 years<br />
Example 2: According to a report, the mean monthly bill for cell phone users in the<br />
United States was $49.70 in 1999. We want to determine whether last year’s mean<br />
monthly bill for cell phone users has decreased from the 1999 mean of $49.70. A<br />
random sample of 50 sample users was selected and average of the last year’s monthly<br />
cell phone bills for these 50 users was $41.40. At the 1% significance level, do we have<br />
enough evidence to conclude that last year’s mean monthly bill for cell phone users has<br />
decreased from the 1999 mean of $49.70 Assume that σ = 25.<br />
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Stat 2601<br />
Example 3: A researcher wishes to find out whether the mean body temperature of<br />
humans is 98.6° F. The researcher obtained the body temperature of 93 healthy humans<br />
and found that the mean body temperature is 98.1°F. At the 1% significance level, do<br />
the data provide sufficient evidence to conclude that the mean body temperature of<br />
healthy humans differs from 98.6°F. Assume that σ = .63°F.<br />
3. P-value Method<br />
1) Definition<br />
P-value- the P-value of a hypothesis test is the probability of observing a value of<br />
the test statistic as extreme or more extreme than that observed when the null<br />
hypothesis is true.<br />
Right-tailed test: The P-value is the probability of observing a value of the test<br />
statistic as large as or larger than the value actually observed, which is the area<br />
under the standard normal curve that lies to the right of the observed test statistic.<br />
Left-tailed test: The P-value is the probability of observing a value of the test<br />
statistic as small as or smaller than the value actually observed.<br />
Two-sided test: The P-value is the probability of observing a value of the test<br />
statistic at least as large in magnitude as the value actually observed.<br />
2) Steps<br />
State the hypotheses.<br />
Compute the value of the test statistic, say z 0 .<br />
Find the P-value.<br />
For an upper-tailed test, P-value = P(z > z 0 ).<br />
For a lower -tailed test, P-value = P(z < z 0 ).<br />
For a two-sided test, P-value = 2*P(z > z 0 ) if z 0 > 0,<br />
or = 2*P(z < z 0 ) if z 0 < 0.<br />
If P –value ≤ α, reject H 0 ; otherwise, do not reject H 0 .<br />
Interpret the results of the hypothesis test.<br />
Example 1: Use the P-value method to perform the same hypothesis test.<br />
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Stat 2601<br />
Example 2 Use the P-value method to perform the same hypothesis test.<br />
Example 3: Use the P-value method to perform the same hypothesis test.<br />
Section 8.3<br />
t Test for a Mean<br />
1. Definition<br />
The t test is a statistical test which can be used when the population is normally<br />
distributed, σ is unknown and n < 30. The test statistic for a t test with null hypothesis<br />
H 0 : μ = μ 0 is:<br />
X<br />
t<br />
0 .<br />
s / n<br />
The degrees of freedom are d.f. = n -1.<br />
2. Finding the rejection region<br />
1) H 1 : µ > µ 0 , rejection region: t > t α .<br />
2) H 1 : µ < µ 0 , rejection region: t t α/2 or t < -t α/2 .<br />
Where t α .is the t-value that will give an area of α to its right. t α .and t α/2 are based<br />
on (n-1) df.<br />
Example: The average undergraduate cost for tuition, fees, and room for two-year<br />
institutions last year was $13,252. The following year, a random sample of 20 two-year<br />
institutions had a mean of $15,560 and a standard deviation of $3500. Is there sufficient<br />
evidence at α = .05 to conclude that the mean cost has increased<br />
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Section 8.4<br />
z Test for a Proportion<br />
Stat 2601<br />
1. The test statistic for a test of hypothesis with H 0 : p = p 0 is:<br />
z<br />
pˆ<br />
p<br />
0<br />
p0<br />
,<br />
q / n<br />
Where pˆ is the sample proportion, n is the sample size, p 0 is the hypothesized value of p.<br />
2. Rejection region (Traditional method)<br />
See Table 8.1.<br />
3. p-value (p-value method)<br />
See Section 8.2.<br />
0<br />
Example 8.4.1: A recent survey found that 64.7% of the population owns their homes. In a<br />
random sample of 150 heads of household, 92 responded that they owned their homes. At the<br />
0.01 level of significance, does that indicate a difference from the national proportion Use<br />
the traditional method.<br />
Example 8.4.2: Nationally 60.2% of federal prisoners are serving time for drug offenses. A<br />
warden feels that in his prison the percentage is even higher. He surveys 400 inmates’<br />
records and finds that 260 of the inmates are drug offenders. At α = 0.05, is he correct Use<br />
the P-value method.<br />
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