Question

1. Suppose you are drawing a random sample of size n > 0 from N(μ, σ2) where σ > 0 is known. Decide if the following statements are true or false and explain your reasoning. Assume our 95% confidence procedure is (X- 1.96X+1.96 Vn a. If (3.2, 5.1) is a 95% CI from a particular random sample, then there is a 95% chance that μ is in this interval. b. If (3.2.5.1) is a 95% CI from a particular random sample, then there is a 95% chance that the mean from our next random sample will be in this interval. C. A 95% CI will contain 95% of the possible values from the population we are studying. d. If we generate 400 random Cls using our 95% confidence procedure, we expect about 20 intervals to not contain μ.

Answer 1

(a) the Statement is TRUE. The significance of 95 % CI is that if we take 100 samples from the population and compute 95% CI for each sample , then approximately 95 out of 100 confidence intervals will contain the true mean. Here (3.2 , 5.1) is 95% CI so there is 95% probability that the mean lies between 3.2 - 5.1 .

(b) The statement is FALSE. In 95% CI, there is some chance for sample mean lies out of the interval and that particular case coming up at any samples. There is no assurance that the very next sample mean must lie on the interval.

(c) The statement is FALSE. CI is defined over parameter. sample values are not parameter , it is sample unit. Here we calculate CI for mean of the population. We take all sample from the given population that is 100% comes from population not 95 %.

(d)The statement is TRUE. in 95% CI , there is a chance of 5% for not containing the true mean. so if we take 400 different intervals for different samples there is a chance of 20 intervals to not contain the true mean.