Question

A sample of 16 individuals is selected from a population with μ = 75. A treatment is administered to the sample. After treatment, M = 82 and s2 = 64.

1. Find the 90% confidence interval.

2. Find the 95% confidence interval.

3. What happens to the width of the interval when you increase the level of confidence (the percent confidence)?

4. Does the confidence interval always include the population mean, sample mean, or both?

Answer 1

Q.1

Q.2

Q 3 . Answer
Pr[ μ- 2 σ < x < μ + 2 σ ] is about 0.68
Pr { μ - 3 σ < x < μ+ 3σ } which is roughly 0.95.
Here we can seen the probability on the right hand side increases, the confidence interval widens and as it decreases, the interval narrows down.
When we increase the probability, naturally the range widens to include more values.
95% confidence interval means, there is a 5% chance of the suggested answer not accurate.
90% confidence interval means, there is a 10% chance of the suggested answer not accurate.
So here chance of being wrong in increases.
Hence increases the level of confidence will be increased the probability of true mean lies in the interval.
Hence the 90% confidence interval is narrower than 95% confidence interval.

Q.4 answer
Confidence interval always include sample mean but doesn't include population mean always.
Here we can seen the confidence interval of 90% and 95% include sample mean but doesn't not include population mean. So we conclude that the population mean is different from true mean.