Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 134
Suppose that 10 sets of hypotheses of the form $$H_{0}: \mu=\mu_{0} \quad H_{1}: \mu \neq \mu_{0}$$ have been tested and that the \(P\) -values for these tests are 0.12 , \(0.08 .0 .93,0.02,0.01,0.05,0.88,0.15,0.13,\) and \(0.06 .\) Use Fisher's procedure to combine all of these \(P\) -values. What conclusions can you draw about these hypotheses?
Short Answer
Expert verified
Combining the P-values with Fisher's method, we can reject the null hypotheses as the combined P-value is significant.
Step by step solution
01
Understand Fisher's Procedure
Fisher's method is used to combine independent P-values from different hypotheses tests. It assumes that each test is independent and follows a chi-squared distribution.
02
Calculate Chi-Square Statistic
For each P-value \( p_i \), calculate \(-2\ln(p_i) \). The combined test statistic is the sum of these values. Given P-values are 0.12, 0.08, 0.93, 0.02, 0.01, 0.05, 0.88, 0.15, 0.13, and 0.06, we compute:\(\begin{align*}-2\ln(0.12) & \approx 4.32, \-2\ln(0.08) & \approx 5.14, \-2\ln(0.93) & \approx 0.14, \-2\ln(0.02) & \approx 7.60, \-2\ln(0.01) & \approx 9.21, \-2\ln(0.05) & \approx 5.99, \-2\ln(0.88) & \approx 0.25, \-2\ln(0.15) & \approx 3.91, \-2\ln(0.13) & \approx 4.32, \-2\ln(0.06) & \approx 5.64.\end{align*}\)
03
Sum Chi-Square Values
Add all the computed values together to get the combined chi-square statistic. Thus, the sum is:\(4.32 + 5.14 + 0.14 + 7.60 + 9.21 + 5.99 + 0.25 + 3.91 + 4.32 + 5.64 = 46.52\)
04
Determine Degrees of Freedom
The degrees of freedom for Fisher's combined test is twice the number of tests (since each \(-2\ln(p_i)\) is approximately chi-squared with 2 degrees of freedom). For 10 tests, the total degrees of freedom is \(10 \times 2 = 20\).
05
Compare with Chi-Square Distribution
Using a chi-squared distribution table or calculator, find the P-value for the combined chi-square statistic, 46.52, with 20 degrees of freedom.
06
Draw Conclusion
If the combined P-value is below a significance level like 0.05, we reject the null hypothesis. Check if 46.52 exceeds the critical value (or check the P-value directly). Here, 46.52 results in a P-value much less than 0.05, suggesting rejecting the null hypothesis of all tests combined.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hypothesis Testing
Hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. It usually involves two hypotheses: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_1 \)). The null hypothesis represents a statement of no effect or no difference, and it's what researchers typically try to disprove. The alternative hypothesis, on the other hand, represents the effect or difference that the researcher wants to prove.
- Step 1 involves stating these hypotheses. For example, if you're comparing the mean of a sample to a known value, you might declare \( H_0: \mu = \mu_0 \) and \( H_1: \mu eq \mu_0 \).
- Then, you collect data and perform a statistical test that calculates a test statistic.
- Finally, determine whether the test statistic falls in the rejection region for the null hypothesis by using a significance level (commonly \( \alpha = 0.05 \) or \( \alpha = 0.01 \)).
Chi-squared Distribution
The chi-squared distribution is a continuous probability distribution that is widely used in hypothesis testing to determine goodness-of-fit, test independence, and to combine probabilities from different tests as in Fisher鈥檚 method. The shape of the chi-squared distribution depends on the degrees of freedom, and it is skewed to the right.
- The goodness-of-fit test uses the chi-squared distribution to see how well a sample fits the expected distribution.
- In hypothesis testing, the chi-squared statistic is often calculated by comparing observed data with data expected under the null hypothesis.
- For combining P-values, as in Fisher's method, we calculate the statistic \(-2 \ln(p_i)\) for each P-value and sum these values.
P-value
The P-value is a crucial concept in hypothesis testing. It helps determine the strength of the results of a test. The P-value represents the probability of observing test results at least as extreme as the ones observed, assuming that the null hypothesis is true.
- A small P-value, typically less than \( 0.05 \), indicates strong evidence against the null hypothesis and suggests that you should reject \( H_0 \).
- Conversely, a large P-value suggests that the observed data is consistent with the null hypothesis, so you would fail to reject \( H_0 \).
- P-values from independent tests can be combined using Fisher's method to evaluate multiple hypotheses simultaneously.
Degrees of Freedom
Degrees of freedom (df) are fundamental in statistics, serving as the number of values in the final calculation of a statistic that are free to vary. In the context of hypothesis testing and specifically the chi-squared distribution, the degrees of freedom determine the exact shape of the chi-squared distribution.
- When combining P-values using Fisher's method, the degrees of freedom for the combined test statistic is twice the number of tests, because each individual test contributes 2 degrees of freedom.
- For example, if you have 10 tests, the combined degrees of freedom would be \( 10 \times 2 = 20 \).
- Degrees of freedom are crucial because they help you identify the appropriate chi-squared distribution curve from which to compare your test statistic.
