The following are some recommended practical self-assessment questions for the lesson called *Better Understanding Statistical Inference by Estimating the Mean and its Confidence Intervals*. They’re intended for you to work through to test your own __ability to apply the key concepts__ we covered there.

Notes:

- You may need to use Excel’s NORMSINV() and TINV() functions for these problems in cases not represented in the printed tables.
- You can assume that the data themselves are characterized by a normal distribution in these examples, i.e. that the population we’re sampling in these problems is normally distributed. This assumption doesn’t matter anyway for large samples (e.g. n >= 30) since the Central Limit Theorem tells us that our approaches to estimating confidence intervals work well for data with any kind of distribution. But if the data are normally distributed then you don’t have to worry for small samples (e.g. n < 30) either – as you might if the distribution of the data itself wasn’t normal.

## Question 1

a) I want you to redo parts a), b), and d) of Example 1 from my notes as if I’d asked for a 68.268949% confidence interval, i.e. pretty much corresponding to -/+ one standard deviation.

b) I want you to redo parts a), b), and d) of Example 1 from my notes as if I’d asked for a 95% confidence interval.

c) I want you to redo parts a), b), and d) of Example 1 from my notes as if I’d asked for a 99.999% confidence interval.

d) What is the width of the confidence interval (upper bound – lower bound) in the original Example 1 where the confidence interval was 99% and in each of the cases in a), b) and c) above?

e) Compare the different confidence intervals achieved in the original Example 1 where the confidence interval was 99% with those in a), b) and c) above. Do this with a sketch showing the intervals along a common axis so we can see the differences. You don’t need to show the normal curves here, but I would like to see the sample mean and the values defining each of the confidence intervals. What I’m after here is you visually representing the differences between the intervals at the different levels of confidence.

f) Comment on the difference between the standard error case and the 99.999% case in terms of what it is you’re confident about in each case.

## Question 2

a) I want you to redo parts a), b) and d) of Example 1 from my notes as if I’d said that N=300. Use a 99% confidence interval as I’d asked for in that original example.

b) What is the width of the confidence interval (upper bound – lower bound) in the original Example 1 where the confidence interval was 99% and the case you saw in a) above?

c) Create a sketch similar to the one you created in Question 1 d).

d) Comment on how sensitive a confidence interval can be to having a finite population.

e) You should have seen a narrowing of the confidence interval. Why is that?

## Question 3

a) I want you to redo parts a), b) and d) of Example 1 from my notes as if I’d said that n=15. Use a 99% confidence interval as I’d asked for in that original example and assume as we had originally in that example that the population is practically infinite in size.

b) I want you to redo parts a), b) and d) of Example 2 from my notes as if I’d said that n=15. Use a 99% confidence interval as I’d asked for in that original example and assume as we had originally in that example that the population is practically infinite in size.

c) What are the widths of the confidence interval (upper bound – lower bound) in the original Example 1 and Example 2 where the population size was practically infinite and in the cases you saw in a) and b) above?

d) Comment on the differences caused by having a sample size smaller than 30. You should have seen a widening of the confidence intervals. Why is that?

e) Comment on the difference between your results on parts a) and b). You should have seen a widening of the width of the interval in b). (And you can even check that it would have widened even if the estimated standard deviation had been the exactly correct value of 3.00 m.) Why is that?

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