Question

What is a rejection region? What is a confidence level? What is a confidence interval? Discuss how the three concepts are related to each other. Provide original examples throughout your discussion

Answer 1

Rejection region:

Rejection region is considered as the critical part of the testing process.

For the hypothesis test (H0) the researcher 1st collects the sample data and next step is to determine the test statistics.To reject the null hypothesis is called the region of rejection (or)

Rejection region: the set of values for the test statistic that leads to rejection of H_o.

EXAMPLE: A researcher might hypothesize that the population mean is equal to 20. To test this null hypothesis, he/she could collect a random sample of observations and compute the sample mean. If the sample mean is close to 20 (say, between 19 and 21), the researcher might decide to accept the hypothesis. In this example, the region of rejection would be the range of values that are less than 19 or greater than 21. If the sample mean falls in this range, the researcher would reject the null hypothesis.

Confidence level:

It is refers to the percentage of all possible samples that can be expected to include the true population parameter.In most cases,

EXAMPLE: the confidence level is taken as 95%,suppose all possible samples were selected from the same population, and a confidence interval were computed for each sample.

Confidence interval:

A range of values such that the population can be expected to contain for the given confidence level is termed as the confidence interval .

(or)  

it is defined as an interval estimate of the population parameter which is calculated for the given data based on the a point estimate & for the given confidence level

RULE:

If the sample size is larger, the value of the sample mean is closer to the population mean.

If the sample size is larger, the value of standard deviation or standard error is smaller.

EXAMPLE:

Say we have a sample mean of 100, standard deviation of 30, and sample size of 50.

For 95% confidence
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    100          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    30          
n = sample size =    50          
              
Thus,              
              
Lower bound =    91.68457705          
Upper bound =    108.3154229          
              
Therefore the confidence interval,   
              
(   91.68457705   ,   108.3154229   )

Rejection region, confidence level, and confidence intervals, are different ways to represent the probability that the observed data is representative of the actual data, or in the case of a rejection region, are not representative of said data.