Section 9'1: Introduction to Statistical Tests Section 9'2: Testing the Mean

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Section 9.1 Introduction to Statistical
TestsSection 9.2 Testing the Mean

• Brase and Brase 9th edition

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Hypothesis testing is used to make decisions
concerning the value of a parameter.
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The value specified in the null hypothesis is
often
Null Hypothesis H0
a working hypothesis about the population
parameter in question

• a historical value
• a claim
• a production specification

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Alternate Hypothesis H1

• any hypothesis that differs from the null
  hypothesis

An alternate hypothesis is constructed in such a
way that it is the one to be accepted when the
null hypothesis must be rejected.
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A manufacturer claims that their light bulbs burn
for an average of 1000 hours. We have reason to
believe that the bulbs do not last that long.

• Determine the null and alternate hypotheses.

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A manufacturer claims that their light bulbs burn
for an average of 1000 hours. ...

• The null hypothesis (the claim) is that the true
  average life is 1000 hours.
• H0 ? 1000

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A manufacturer claims that their light bulbs
burn for an average of 1000 hours. We have reason
to believe that the bulbs do not last that long.
...
If we reject the manufacturers claim, we must
accept the alternate hypothesis that the light
bulbs do not last as long as 1000 hours. H1 lt
1000
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Type I Error

• rejecting a null hypothesis which is, in fact,
  true

Type II Error
not rejecting a null hypothesis which is, in
fact, false
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Errors in Hypothesis Testing
Our Choices
Correct decision
Type I error
H0 is
Correct decision
Type II error
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Level of Significance, Alpha (?)

• the probability with which we are willing to risk
  a type I error
• (rejecting H0 when it is true)

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Type II Error
? beta probability of a type II error
(failing to reject a false hypothesis) A small
? is normally is associated with a (relatively)
large ?, and vice-versa. Choices should be made
according to which error is more serious.
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Power of the Test 1 Beta

• The probability of rejecting H0 when it is in
  fact false 1 ?.
• The power of the test increases as the level of
  significance (?) increases.
• Using a larger value of alpha increases the power
  of the test but also increases the probability of
  rejecting a true hypothesis.

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Probabilities Associated with a Hypothesis Test
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Reject or ...

• When the sample evidence is not strong enough to
  justify rejection of the null hypothesis, we fail
  to reject the null hypothesis.
• Use of the term accept the null hypothesis
  should be avoided.
• When the null hypothesis cannot be rejected, a
  confidence interval is frequently used to give a
  range of possible values for the parameter.

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Fail to Reject H0

• There is not enough evidence to reject H0. The
  null hypothesis is retained but has not been
  proven.

Reject H0
There is enough evidence to reject H0. Choose
the alternate hypothesis with the understanding
that it has not been proven.
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Statistical Significance

• If we reject H0, we say that the data collected
  in the hypothesis testing process are
  statistically significant.
• If we do not reject H0, we say that the data
  collected in the hypothesis testing process are
  not statistically significant.

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A fast food restaurant indicated that the average
age of its job applicants is fifteen years. We
suspect that the true age is lower than 15. We
wish to test the claim with a level of
significance of ? 0.01,
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average age of its job applicants is fifteen
years. We suspect that the true age is lower
than 15.

• H0 ? 15
• H1 ? lt 15
• Describe Type I and Type II errors.

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Errors in Hypothesis Testing
Our Choices
Correct decision
Type I error
H0 is
Correct decision
Type II error
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H0 ? 15 H1 ? lt 15 ? 0.01

• A type I error would occur if we rejected the
  claim that the mean age was 15, when in fact the
  mean age was 15 (or higher). The probability of
  committing such an error is as much as 1.

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H0 ? 15 H1 ? lt 15 ? 0.01
A type II error would occur if we failed to
reject the claim that the mean age was 15, when
in fact the mean age was lower than 15. The
probability of committing such an error is called
beta.
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P-value

• Assuming H0 is true, the probability that the
  test statistic will take on values as extreme as
  or more extreme than the observed test statistic
  (computed from sample data) is called the P-value
  of the test. The smaller the P-value computed
  from the sample data, the stronger the evidence
  against H0.

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Test Statistic

• Given that x has a normal distribution with known
  standard deviation s, then
• Test statistic

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Two-Tailed Test
H0 ? k H1 ? ? k
P-value2P(zgtzx)
If test statistic is at or near the claimed mean,
we do not reject the Null Hypothesis
z 0 z
If test statistic is in either tail - the
critical region - of the distribution, we reject
the Null Hypothesis.
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Right-Tailed Test
P-valueP(zgtzx)
H0 ? k H1 ? gt k
If test statistic is at, near, or below the
claimed mean, we do not reject the Null Hypothesis
0 z
If test statistic is in the right tail - the
critical region - of the distribution, we reject
the Null Hypothesis.
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Left-Tailed Test
P-valueP(zltzx)
H0 ? k H1 ? lt k
If test statistic is at, near, or above the
claimed mean, we do not reject the Null Hypothesis
z 0
If test statistic is in the left tail - the
critical region - of the distribution, we reject
the Null Hypothesis.
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How to conclude a test using the P-value and
level of significance

• If P-valuelta we reject the null hypothesis.
• If P-valuegta, we do not reject the null
  hypothesis.

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Basic components of a statistical test

• The null hypothesis, H0, the alternate
• hypothesis, H1, and level of significance.
• 2. Test statistic and sampling distribution.
• 3. P-value.
• 4. Test conclusion.
• 5. Interpretation of the test results.

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How to test m when s is known

• State H0 and H1 and the level of significance a.
• If you can assume that x has a normal
  distribution, then any n will work, otherwise
  ngt30. Use the know s, the sample size n, the
  value of from the sample, and m from the null
  hypothesis to compute the standardized sample
  test statistic.

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How to test m when s is known

• 3. Use the standard distribution and the type of
  test, one-tailed or two-tailed, to find the
  P-value corresponding to the test statistic.
• 4. If P-valuelta, the reject H0. If P-valuegta,
  the do not reject H0.
• 5. State your conclusion in the context of the
  application.

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Example Sun Spots

• X represents the number of sun spots observed in
  a four week period.
• s35
• n40
• a.05
• H0 m41 and H1 mgt41
• 2.

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3. Find the P-value Right-Tailed Test ? 0.05
P-valueP(zgt1.08)1-.8599.1401
H0 ? 41 H1 ? gt 41
P-value
0 1.08
Sample test statistic
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Example

• 4. Since the P-value.1401gt.05 for a, we do not
  reject H0.
• 5. Based on the sample data we do not think the
  average sunspot activity during the sample time
  period was higher than the long-term mean.