(DOE) -

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2 Simple Comparative Experiments

• Statistical Plots
• Sampling and Sampling Distributions
• Hypothesis Testing
• Confidence Interval

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?? (Dot Diagram)

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??? (Histogram)
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??? (Box Plot)
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????? (Time Series Plot)
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??????????

• ?????(m ) ???????? E(X)
• ?????(s 2) ???????? V(X)

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Sample and Sampling
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???(Point Estimation)

• ??????????, ?????????????, ???????, ?????.
• ??????????
• ???
• ????
• ??????
• ?????(m)
• ?????(s 2)

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Central Limit Theorem
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????(Hypothesis Testing)

• A person is innocent until proven guilty beyond
  a reasonable doubt. ??????????????, ????????.
• ????
• H0 m 50 cm/s
• H1 m ? 50 cm/s
• Null Hypothesis (H0) Vs. Alternative Hypothesis
  (H1)
• One-sided and two-sided Hypotheses
• A statistical hypothesis is a statement about the
  parameters of one or more populations.

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About Testing

• Critical Region
• Acceptance Region
• Critical Values

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Errors in Hypothesis Testing

• ???????
• Type I Error(a) Reject H0 while H0 is true.
• Type II Error(b) Fail to reject H0 while H0 is
  false.

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Making Conclusions

• We always know the risk of rejecting H0, i.e., a,
  the significant level or the risk.
• We therefore do not know the probability of
  committing a type II error (b).
• Two ways of making conclusion
• 1. Reject H0
• 2. Fail to reject H0, (Do not say accept H0)
• or there is not enough evidence to reject
  H0.

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Significant Level (a)

• a P(type I error) P(reject H0 while H0 is
  true)

n 10, s 2.5 s/?n 0.79
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The Power of a Statistical Test

• Power 1 - b
• Power the sensitivity of a statistical test

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General Procedure for Hypothesis Testing
1. From the problem context, identify the
parameter of interest. 2. State the null
hypothesis, H0. 3. Specify an appropriate
alternative hypothesis, H1. 4. Choose a
significance level a. 5. State an appropriate
test statistic. 6. State the rejection region for
the statistic. 7. Compute any necessary sample
quantities, substitute these into the
equation for the test statistic, and compute
that value. 8. Decide whether or not H0 should be
rejected and report that in the problem
context.
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Inference on the Mean of a Population-Variance
Known

• H0 m m0
• H1 m ? m0 , where m0 is a specified constant.
  
• Sample mean is the unbiased point estimator for
  population mean.

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Example 8-2
Aircrew escape systems are powered by a solid
propellant. The burning rate of this propellant
is an important product characteristic.
Specifications require that the mean burning rate
must be 50 cm/s. We know that the standard
deviation of burning rate is 2 cm/s. The
experimenter decides to specify a type I error
probability or significance level of a 0.05.
He selects a random sample of n 25 and obtains
a sample average of the burning rate of x 51.3
cm/s. What conclusions should be drawn?
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1. The parameter of interest is m, the meaning
   burning rate.
2. H0 m 50 cm/s
3. H1 m ? 50 cm/s
4. a 0.05
5. The test statistics is
6. Reject H0 if Z0 gt 1.96 or Z0 lt -1.96 (because
   Za/2 Z0.025 1.96)
7. Computations
8. Conclusions Since Z0 3.25 gt 1.96, we reject
   H0 m 50 at the 0.05 level of significance. We
   conclude that the mean burning rate differs from
   50 cm/s, based on a sample of 25 measurements.
   In fact, there is string evidence that the mean
   burning rate exceeds 50 cm/s.

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P-Values in Hypothesis Tests

• Where Z0 is the test statistic, and ?(z) is the
  standard normal cumulative function.

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The Sample Size (I)

• Given values of a and d, find the required sample
  size n to achieve a particular level of b..

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The Operating Characteristic Curves- Normal test
(z-test)

• Use to performing sample size or type II error
  calculations.
• The parameter d is defined as
• so that it can be used for all problems
  regardless of the values of m0 and s.
• ??41????????????????????????

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Construction of the C.I.

• From Central Limit Theory,

• Use standardization and the properties of Z,

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Inference on the Mean of a Population-Variance
Unknown

• Let X1, X2, , Xn be a random sample for a normal
  distribution with unknown mean m and unknown
  variance s2. The quantity
• has a t distribution with n - 1 degrees of
  freedom.

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Inference on the Mean of a Population-Variance
Unknown

• H0 m m0
• H1 m ? m0 , where m0 is a specified constant.
  
• Variance unknown, therefore, use s instead of s
  in the test statistic.

• If n is large enough (? 30), we can use Z-test.
  However, n is usually small. In this case, T0
  will not follow the standard normal distribution.

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Inference for the Difference in Means-Two Normal
Distributions and Variance Unknown

• Why?

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is distributed approximately as t with
degrees of freedom given by
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C.I. on the Difference in Means
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C.I. on the Difference in Means
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Paired t-Test

• When the observations on the two populations of
  interest are collected in pairs.
• Let (X11, X21), (X12, X22), , (X1n, X2n) be a
  set of n paired observations, in which X1j(m1,
  s12) and X2j(m2, s22) and Dj X1j X2j, j 1,
  2, , n. Then, to test H0 m1 m2 is the same
  as performing a one-sample t-test H0 mD 0
  since
• mD E(X1-X2) E(X1)-E(X2) m1 - m2

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Inference on the Variance of a Normal Population
(I)

• H0 s2 s02
• H1 s2 ? s02 , where s02 is a specified
  constant.
• Sampling from a normal distribution with unknown
  mean m and unknown variance s2, the quantity
• has a Chi-square distribution with n-1 degrees
  of freedom. That is,

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Inference on the Variance of a Normal Population
(II)

• Let X1, X2, , Xn be a random sample for a normal
  distribution with unknown mean m and unknown
  variance s2. To test the hypothesis
• H0 s2 s02
• H1 s2 ? s02 , where s02 is a specified
  constant.
• We use the statistic
• If H0 is true, then the statistic has a
  chi-square distribution with n-1 d.f..

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The Reasoning

• For H0 to be true, the value of ?02 can not be
  too large or too small.
• What values of ?02 should we reject H0? (based on
  a value)
• What values of ?02 should we conclude that
  there is not enough evidence to reject H0?

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Example 8-11

• An automatic filling machine is used to fill
  bottles with liquid detergent. A random sample
  of 20 bottles results in a sample variance of
  fill volume of s2 0.0153 (fluid ounces)2. If
  the variance of fill volume exceeds 0.01 (fluid
  ounces)2, an unacceptable proportion of bottles
  will be underfilled and overfilled. Is there
  evidence in the sample data to suggest that the
  manufacturer has a problem with underfilled and
  overfilled bottles? Use a 0.05, and assume
  that fill volume has a normal distribution.

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Hypothesis Testing on Variance - Normal
Population
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The Test Procedure for Two Variances Comparison
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Hypothesis Testing on the Ratio of Two Variances