Question

1. In order to construct a confidence interval for a population mean or proportion, we must have that the sample size is small relative to the population and the sample data must have come from a simple random sample or a randomized experiment. Explain why these conditions are necessary.

2. In order to construct a confidence interval for a population proportion, we must additionally have np(1-p) at least 10. For a population mean, we must additionally have that the population is normally distributed or the sample size is greater than or equal to 30. Explain why these conditions are necessary.

3. Discuss some of the benefits and drawbacks of constructing a confidence interval for a high confidence level, such as 99%. Discuss some of the benefits and drawbacks of using a low confidence level, such as 90%.

Answer 1

There are certain conditions to use confidence interval which are :-

1) Randomization Condition

2) Independence Condition

3) 10% Condition

4) Sample size condition

a) The sample size must come from the random sample because for using the confidence interval the assumptions of central limit theorem should be satisfied and randomization is one of the assumptions. With the smaller sample as relative to population, the confidence interval get narrower. So, it will be useful.

b) While using the proportio, np and n(1-p) should be at least greater than 10. As we have to check that the normality assumption is satisfied i.e. the sample size is large and probability of success/ failure is very small. As we know we can use confidence interval when the assumption of central limit is met. It is met when the sample size is large enought to met normality criteria. In general we took sample size as greater than 30, which is considered to be sufficient.

c) The benefits of constructing a confidence interval of 99% is that there will be approximately 99 of the 100 confidence interval will contain the parameter. But here we have to compromise with precision. As the confidence interval will be wider.

But in constructing a confidence interval of 90% is that there will be approximately 90 of the 100 confidence interval will contain the parameter which is lesser as compare to 99% .  But here we will met with the precision. As the confidence interval will be narrower.