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MANE 3332.03

Lecture 24, April 22

  • Topics:

    • Chapter 9, Case 2
    • Errors in Hypothesis Testing

  • Assignments:

    • New: Chapter 9, Case 1 Upper Quiz (assigned 4/22/2025, due 4/24/2025)
    • New: Chapter 9, Case 1 Lower Quiz (assigned 4/22/2025, due 4/24/2025)
    • New: Chapter 9, Case 1 2-sided Quiz (assigned 4/22/2025, due 4/23/2025)
    • New: Chapter 9, Case 2 Upper Practice Problems (assigned 4/22/2025, due 4/24/2025)
    • New: Chapter 9, Case 2 Lower Practice Problems (assigned 4/22/2025, due 4/24/2025)
    • New: Chapter 9, Case 2 2-sided Practice Problems (assigned 4/22/2025, due 4/23/2025)
    • New: Technical Report Two (assigned 4/17/2025, due 5/15/2025)

  • Attendance

  • Questions?

Handouts

Class Schedule

Tuesday Lecture Thursday Lecture
4/22: Chapter 9, Case 2 4/24: Chapter 9, Case 3
4/29: Chapter 11 5/1: Chapter 11
5/6: Review 5/8: Dead Day (no class)

5 Sessions plus final exam

Final Exam: Tuesday May 13, 2025 10:15 am - 12:00 pm

Errors in hypothesis testing

  • Whether a correct decision is made depends upon the true nature of \(H_0\) and the decision arrived at.

  • A type I error occurs when the null hypothesis is true and the outcome of the test is to reject \(H_0\). The probability of a type I error is denoted as \(\alpha\)

  • A type II error occurs when the null hypothesis is false and the outcome of the test is to fail to reject \(H_0\). The probability of a type II error is denoted as \(\beta\).

  • The power of a statistical test is the probability rejecting the null hypothesis \(H_0\) when the alternative hypothesis is true. \(\mbox{Power}=1-\beta\)

Error Example: Manufacturing

Error Example: Medical

Chapter 9, Case 1 - Type II error and sample size for a two-tailed test

  • Probability of type II error for the two-tailed test
\[ \beta=\Phi\left(z_{\alpha/2}-\frac{\delta\sqrt{n}}{\sigma}\right)-\Phi\left(-z_{\alpha/2}-\frac{\delta\sqrt{n}}{\sigma}\right) \]

where \(\mu=\mu_0+\delta\)

  • The sample to detect a difference between the true and hypothesized mean of \(\delta\) with power at least \(1-\beta\) is
\[ n\approx\frac{(z_{\alpha/2}+z_\beta)^2\sigma^2}{\delta^2} \]

where \(\delta=\mu-\mu_0\)

Type II error and sample size for the one-tailed tests

  • For an upper-tailed test
\[ \beta=\Phi\left(z_\alpha-\frac{\delta\sqrt{n}}{\sigma}\right) \]
  • For a lower-tailed test
\[ \beta=1-\Phi\left(-z_\alpha-\frac{\delta\sqrt{n}}{\sigma}\right) \]
  • The sample size required to detect a difference between the true mean and hypothesized mean of \(\delta\) with power at least \(1-\beta\) is
\[ n=\frac{(z_\alpha+z_\beta)^2\sigma^2}{\delta^2} \]

If \(n\) is not an integer, round up to the nearest integer

R: Chapter 9 Case 1 Hypothesis Testing

Power using OC-Curve

Find the power when the true mean value is 3.325

R: Chapter 9 Case 1 Power

Type II Error Rate and Sample Size

  • You will not be required to calculate or use OC-curves
  • You must understand the concept and be able to correctly identity type I and type II error

Chapter 9, Case 2

Hypothesis Test on the Mean, Variance Unknown

  • Much more common case than variance known

  • Substitute \(S\) for \(\sigma\)

  • The test statistics is now a \(t\) random variable

\[ T=\frac{\overline{X}-\mu_0}{S/\sqrt{n}} \]

Summary of Case 2

Problem 9.3.6

Classical Approach

Hypothesis Test Using R

Normal Probability Plot

\(P\)-values

  • More difficult to calculate since the \(t\)-tables only contain a few quantiles

  • Can use tables to generate bounds on the \(p\)-value

  • Software will provide \(p\)-values

P-values from R

Power Calculations

  • Are much more complicated

  • The true distribution is now a non-central \(t\)

  • Use tables to solve (Chart VII in appendix) or software

Power Calculation using R

Sample Size using R

Chapter 9, Case 2 2-sided Practice Problems

Chapter 9, Case 2 Lower Practice Problems

Chapter 9, Case 2 Upper Practice Problems

Case 3. Hypothesis Test on Variance of Normal Population

  • The test statistics is a \(\chi^2\) random variable
\[ \chi^2_0=\frac{(n-1)S^2}{\sigma^2_0} \]

- The table below summarizes the three possible hypothesis tests. The rejection regions are clearly shown in Figure 9-17 on page 222

Figure 9-17

Test Summary

  • See summary in your textbook

Problem 7.108

Problem taken from Ostle, Turner, Hicks and McElrath (1996). Engineering Statistics: The Industrial Experience. Duxbury Press.

Statistics for Problem 7.108

Classical Approach

Problem 7.108 Plots

Problem 7.108 Plots

Problem 7.108 Shapiro-Wilks Test

p-values

  • Very similar to the case for the mean of a normal population with variance unknown

  • Difficult to calculate since the \(\chi^2\)-tables only contain a few quantiles

  • Can use tables to generate bounds on the \(p\)-value

  • Software will provide \(p\)-values

Test on Variance using R (EnvStats)

Power Calculations

  • Can be done with OC curves found in Table VIIi--n

  • Can be done in software such as R

Test on Standard Deviation

  • What about test on standard deviation?

Chapter 9, Case 3 2-sided Practice Problems

Practice Problem Chapter 9, Case 3 Lower

Practice Problem Chapter 9, Case 3 Upper

Case 4. Hypothesis Test on a Population Proportion

  • The test statistics for the hypothesis test is
\[ Z_0=\frac{x-np_0}{\sqrt{np_0(1-p_0)}} \]

Problem 9.5.2

Problem 9.5.2 Classic Approach

Power Calculations

  • For the two-sided alternative hypothesis
\[ \begin{aligned} \beta&=&\Phi\left(\frac{p_0-p+z_{\alpha/2}\sqrt{p_0(1-p_0)/n}}{\sqrt{p(1-p)/n}}\right)\\ &&-\Phi\left(\frac{p_0-p-z_{\alpha/2}\sqrt{p_0(1-p_0)/n}}{\sqrt{p(1-p)/n}}\right) \end{aligned} \]
  • If the alternative is \(H_1:p<p_0\)
\[ \beta=1- \Phi\left(\frac{p_0-p-z_{\alpha}\sqrt{p_0(1-p_0)/n}}{\sqrt{p(1-p)/n}}\right) \]
  • and finally if the alternative hypothesis is \(H_1:p>p_0\)
\[ \beta=\Phi\left(\frac{p_0-p+z_{\alpha}\sqrt{p_0(1-p_0)/n}}{\sqrt{p(1-p)/n}}\right) \]

Sample Size

  • Sample size requirements to satisfy type II(\(\beta\)) error constraints for a two-tailed hypothesis test is given by
\[ n=\left[\frac{z_{\alpha/2}\sqrt{p_0(1-p_0)}+z_\beta\sqrt{p(1-p)}}{p-p_0}\right]^2. \]

  • For a sample size for a one-sided test substitute \(z_\alpha\) for \(z_{\alpha/2}\).

  • Problem 9.95

Testing for Goodness of Fit

  • Material is presented in section 9-7 of your textbook

  • Procedure determines if the sample data is from a specified underlying distribution

  • Procedure uses a \(\chi^2\) distribution

  • Example 9-12 presents a \(\chi^2\) goodness of fit test for a Poisson example

  • Example 9-13 presents a \(\chi^2\) goodness of fit test for a normal example

Procedure

  1. Collect a random sample of size \(n\) from a population with an unknown distribution,

  2. Arrange the \(n\) observations in a frequency distribution containing \(k\) classes

  3. Calculate the observed frequency in each class \(O_i\),

  4. From the hypothesized distribution, calculate the expected frequency in class \(i\), denoted \(E_i\) (if \(E_i\) is small combine classes)

  5. Calculate the test statistic

\[ \chi^2_0=\frac{\sum_{i=1}^k\left(O_i-E_i\right)^2}{E_i} \]
  1. Reject the null hypothesis if the calculated value of the test statistic \(\chi^2_0>\chi^2_{\alpha,k-p-1}\) where \(p\) is the number of parameters in the hypothesized distribution

Example 9.12, part 1

Example 9.12, part 2

Chapter 9 Summary

  • You should be prepared to work any practice problems assigned: Cases 1-3 with three different alternatives
  • All other information is conceptual knowledge that can be questioned with multiple choice
    • Name 3 ways to test if data is from a normal distribution