Inference on the Mean of a Population -Variance Known

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Inference on the Mean of a Population-Variance
Known
4-4 (8-2)

• 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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The Reasoning

• For H0 to be true, the value of Z0 can not be too
  large or too small.
• Recall that 68.3 of Z0 should fall within (-1,
  1)
• 95.4 of Z0 should fall within (-2, 2)
• 99.7 of Z0 should fall within (-3, 3)
• What values of Z0 should we reject H0? (based on
  a value)
• What values of Z0 should we conclude that there
  is not enough evidence to reject H0?

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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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Hypothesis Testing on m - Variance Known
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P-Values in Hypothesis Tests(I)

• Where Z0 is the test statistic, and ?(z) is the
  standard normal cumulative function.
• In example 8-2, Z0 3.25, P-Value 21-F(3.25)
  0.0012

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P-Values of Hypothesis Testing on m - Variance
Known
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P-Values in Hypothesis Tests(II)

• a-value is the maximum type I error allowed,
  while P-value is the real type I error calculated
  from the sample.
• a-value is preset, while P-value is calculated
  from the sample.
• When P-value is less than a-value, we can safely
  make the conclusion Reject H0. By doing so,
  the error we are subjected to (P-value) is less
  than the maximum error allowed (a-value).

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Type II Error- Fail to reject H0 while H0 is
false
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How to calculate Type II Error? (I)(H0 m m0
Vs. H1 m ? m0)

• Under the circumstance of type II error, H0 is
  false. Supposed that the true value of the mean
  is m m0 d, where ? gt 0. The distribution of
  Z0 is

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How to calculate Type II Error? (II) - refer to
section 4.3 (8.1)

• Type II error occurred when (fail to reject H0
  while H0 is false)

• Therefore,

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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 Sample Size (II)

• Two-sided Hypothesis Testing

• One-sided Hypothesis Testing

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Example 8-3
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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.
• Chart VI a,b,c,d are for Z-test.

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Example 8-5
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Large Sample Test

• If n ? 30, then the sample variance s2 will be
  close to s2 for most samples.
• Therefore, if population variance s2 is unknown
  but n ? 30, we can substitute s with s in the
  test procedure with little harmful effect.

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Large Sample Hypothesis Testing on m - Variance
Unknown but n ? 30
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Statistical Vs. Practical Significance

• Practical Significance 50.5-50 0.5
• Statistical Significance P-Value for each sample
  size n.

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Notes

• be careful when interpreting the results from
  hypothesis testing when the sample size is large,
  because any small departure from the hypothesized
  value m0 will probably be detected, even when the
  difference is of little or no practical
  significance.
• In general, two types of conclusion can be drawn
• 1. At a 0., we have enough evidence to
  reject H0.
• 2. At a 0., we do not have enough evidence
  to reject H0.

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Confidence Interval on the Mean (I)

• Point Vs. Interval Estimation
• The general form of interval estimate is
• L ? m ? U
• in which we always attach a possible error a
  such that
• P(L ? m ? U) 1-a
• That is, we have 1-a confidence that the true
  value of m will fall within L, U.
• Interval Estimate is also called Confidence
  Interval (C.I.).

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Confidence Interval on the Mean (II)

• L is called the lower-confidence limit and
• U is the upper-confidence limit.
• Two-sided C.I. Vs. One-sided C.I.

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Construction of the C.I.

• From Central Limit Theory,

• Use standardization and the properties of Z,

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Formula for C.I. on the Mean with Variance Known

• Used when
• 1. Variance known
• 2. n ? 30, use s to estimate s.

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Example 8-6 Consider the rocket propellant
problem in Example 8-2. Find a 95 C.I. on the
mean burning rate?

• 95 C.I gt a 0.05,
• za/2 z0.025 1.96

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Notes - C.I.

• Relationship between Hypothesis Testing and C.I.s
• Confidence level (1-a) and precision of
  estimation (C.I. 1/2)
• Sample size and C.I.s

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Choice of Sample Size to Achieve Precision of
Estimation
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Example 8-7
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One-Sided C.I.s on the Mean
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Inference on the Mean of a Population-Variance
Unknown
4-5 (8-3)

• 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 the test
  procedure in 4-4 (8-2). However, n is usually
  small. In this case, T0 will not follow the
  standard normal distribution.

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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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The Reasoning

• For H0 to be true, the value of T0 can not be too
  large or too small.
• What values of T0 should we reject H0? (based on
  a value)
• What values of T0 should we conclude that there
  is not enough evidence to reject H0?
• Although when n ? 30, we can use Z0 in section
  8-2 to perform the testing instead. We prefer
  using T0 to more accurately reflect the real
  testing result if t-table is available.

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Example 8-8
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Testing for Normality (Example 8-8)- t-test
assumes that the data are a random sample from a
normal population

• (1) Box Plot (2) Normality Probability Plot

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Hypothesis Testing on m - Variance Unknown
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Finding P-Values

• Steps
• 1. Find the degrees of freedom (k n-1)in the
  t-table.
• 2. Compare T0 to the values in that row and find
  the closest one.
• 3. Look the a value associated with the one you
  pick. The p-value of your test is equal to this
  a value.
• In example 8-8, T0 4.90, k n-1 21, P-Value
  lt 0.0005 because the t value associated with (k
  21, a 0.0005) is 3.819.

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P-Values of Hypothesis Testing on m - Variance
Unknown
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The Operating Characteristic Curves- t-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.
• Chart VI e,f,g,h are used in t-test. (pp.
  A14-A15)

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Example 8-9

• In example 8-8, if the mean load at failure
  differs from 10 MPa by as much as 1 MPa, is the
  sample size n 22 adequate to ensure that H0
  will be rejected with probability at least 0.8?
• s 3.55, therefore, d 1.0/3.55 0.28.
• Appendix Chart VI g, for d 0.28, n 22 gt b
  0.68
• The probability of rejecting H0 m 10 if the
  true mean exceeds this by 1.0 MPa (reject H0
  while H0 is false) is approximately 1 - b 0.32,
  which is too small. Therefore n 22 is not
  enough.
• At the same chart, d 0.28, b 0.2 (1-b0.8)
  gt n 75

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Construction of the C.I. on the Mean - Variance
Unknown

• In general, the distribution of
• is t with n-1 d.f.

• Use the properties of t with n-1 d.f.,

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Formula for C.I. on the Mean with Variance Unknown
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Example 8-10 Reconsider the tensile adhesive
problem in Example 8-8. Find a 95 C.I. on the
mean?

• N 22, sample mean 13.71, s 3.55, ta/2,n-1
  t0.025,21 2.080
• 13.71 - 2.080 (3.55) / ?22 ? m ? 13.71 2.080
  (3.55) / ?22
• 13.71 - 1.57 ? m ? 13.71 1.57
• 12.14 ? m ? 15.28
• The 95 C.I. On the mean is 12.14, 15.28

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Final Note for the Inference on the Mean