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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:

1. You may need to use Excel’s NORMSINV() and TINV() functions for these problems in cases not represented in the printed tables.
2. 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.

#### Numerical solutions to Question 1

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## 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?

#### Numerical solutions to Question 2

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## 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?

#### Numerical solutions to Question 3

Like everything on this site, these solutions are freely accessible. You just need to be signed in »

Welcome (back)!

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If you’re one of my students, then you’re expected to answer these on your own and submit them according to the directions provided in class, i.e. you don’t  need to submit them through this website. Don’t forget that our TA and I are both here to help you in the associated lab (and/or tutorial) sessions.

Aim to provide succinct answers to these applications questions in the same document you created for the conceptual self-assessment questions.

I’d also like you to submit an Excel spreadsheet along with that document. This will likely require a bit of thoughtful organization on your part. For example, It would work well to put the answers to both of the above application problems onto their own sheet / tab within your spreadsheet, and name it accordingly. And then you can refer to that specific tab from your written document.

When you answer other applied question sets in future topics, you should do the calculations on separate sheets / tabs, also appropriately named. And refer to them where required to show your work. This way, you’ll have all of your Excel work in one nicely indexed place and will only need to hand in a single spreadsheet as an appendix to each set of self-assessments.

You can click through to other self-assessments or lessons (if any) using the button below, and return here whenever you wish.