Testing Hypotheses with Bernoulli Distribution

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SUMMARY

This discussion focuses on hypothesis testing using the Bernoulli Distribution with unknown parameter P. The null hypothesis (Ho: P=Po) and alternative hypothesis (H1: P=P1) are tested by determining a critical value c, where the rejection of Ho occurs when Xbar < c. The discussion emphasizes calculating the probabilities α(δ) and β(δ) to establish the test's size and suggests using the normal approximation for large sample sizes based on the De Moivre-Laplace limit theorem.

PREREQUISITES
  • Understanding of Bernoulli Distribution and its parameters
  • Knowledge of hypothesis testing concepts
  • Familiarity with Binomial distribution and its properties
  • Basic statistics, including concepts of α (Type I error) and β (Type II error)
NEXT STEPS
  • Study the derivation of α(δ) and β(δ) in hypothesis testing
  • Learn about the application of the De Moivre-Laplace limit theorem in statistical analysis
  • Explore randomized tests for hypothesis testing
  • Investigate the use of normal approximation in Binomial distributions for large sample sizes
USEFUL FOR

Statisticians, data analysts, and researchers involved in hypothesis testing and statistical inference, particularly those working with Bernoulli and Binomial distributions.

LBJking123
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This is the question:
Suppose that X1...Xn form a random sample from the Bernoulli Distribution with unknown parameter P. Let Po and P1 be specified values such that 0<P1<Po<1, and suppose that is desired to test the following simple hypotheses: Ho: P=Po, H1: P=P1.
A. Show that a test procedure for which α(δ) + β(δ) is a minimum rejects Ho when Xbar < c.
B. Find the value of c.

I know that this problem is not that difficult I just can't figure out where to start. I know the Bernoulli distribution, but I can't figure out how to get α(δ) and β(δ). I have not seen any problems like this so I am kinda lost, any help would be much appreciated. Thanks!
 
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You didn't say what \delta<br /> is.
 
ƩX ~ Binomial (n,p0) under null hyp (H). You know the probabilities of X=0,1, ...,n under H. Find the value of X (= c, say) such that P[X≤c-1] < α ≤ P[X≤c]. Now compare your observed value of X with c.
If you want a test of exact size α, then a randomized test is to be done.
For large n you can use normal approximation due to De Moivre Laplace limit theorem.
 
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