Bernoulli confidence intervals

In summary, the conversation discusses using the Central Limit Theorem (CLT) to construct an approximate symmetric confidence interval for a parameter p. It is suggested to use the error function to solve the problem, but the question remains how to formally prove the result using the cumulative distribution function (CDF).
  • #1
Gekko
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Confidence intervals

1. Homework Statement [/

Use CLT to construct approximate symmetric 100(1-alpha)% confidence interval [L,R] for p then show that [L,1] is then an approximate 100(1-alpha/2)% confidence interval for p


The Attempt at a Solution




When [L,1] then we have a one sided confidence interval.
What we effectively need to show is that the area under the normal curve from -inf to alpha/2 is equal to the area under the curve from -inf to alpha/4 + the area under the curve from alpha/4 to inf
I was going to look at the error function as a way to solve this. I haven't managed it though. Any thoughts?
 
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  • #2
This isn't related to Bernoulli. Thanks to CLT we assume normality so the question is how to formally prove the above in a mathematical way.
Can we just use the CDF?
 

Related to Bernoulli confidence intervals

1. What is a Bernoulli confidence interval?

A Bernoulli confidence interval is a statistical technique used to estimate the true proportion or probability of a binary outcome (such as success or failure) based on a sample of data. It takes into account the variability and uncertainty of the data to provide a range of values within which the true proportion is likely to fall with a certain level of confidence.

2. How is a Bernoulli confidence interval calculated?

A Bernoulli confidence interval is typically calculated using the formula p ± z*√(p(1-p)/n), where p is the observed proportion in the sample, z is the critical value from the standard normal distribution corresponding to the desired confidence level, and n is the sample size. This formula assumes that the sample is representative of the population and that the observations are independent and identically distributed.

3. What is the significance of the confidence level in a Bernoulli confidence interval?

The confidence level in a Bernoulli confidence interval represents the percentage of times that the true proportion is expected to fall within the calculated interval. For example, a 95% confidence level means that if the same sample were taken multiple times, the true proportion would be expected to fall within the calculated interval 95% of the time. Higher confidence levels indicate a greater degree of certainty in the estimated interval.

4. What are the limitations of Bernoulli confidence intervals?

One of the main limitations of Bernoulli confidence intervals is that they assume a fixed sample size and a large enough sample to meet the requirements of the central limit theorem. This means that they may not be appropriate for small sample sizes or when the proportion of interest is very close to 0 or 1. Additionally, they only provide an estimate of the population proportion and cannot be used to make causal inferences.

5. How can Bernoulli confidence intervals be used in scientific research?

Bernoulli confidence intervals can be used in scientific research to estimate the proportion of a population that possesses a certain characteristic or experiences a certain outcome. They can also be used to compare proportions between different groups or to assess the effectiveness of an intervention. However, it is important to understand their limitations and to interpret the results in the context of the specific research question and study design.

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