Statistics II – Interval Estimation: Exercises

This is a set of exercises prepared by the teaching faculty for Statistics II – Applied Statistics for Finance, covering Interval Estimation (confidence intervals for the mean, proportion, and variance), from the Lisbon Accounting and Business School (ISCAL–IPL).

The workbook consists of multiple-choice questions — for which you must select the correct option and justify your choice — and True (T) or False (F) questions — where you must indicate the logical value of each statement and justify your answer adequately.

Exercises

Exercise 1

Consider a population with a Normal distribution and a known standard deviation of \(\sigma = 20\) monetary units (m.u.). A random sample of size \(n = 20\) was drawn, yielding a sample mean of \(\bar{x} = 320\) m.u.

1.1. A 90% confidence interval for the population mean is:

1.2. The minimum sample size needed so that the width of the corresponding 90% confidence interval for the mean does not exceed 1 m.u. is \(n \geq 4{,}330\).

1.3. The minimum sample size needed so that the width of the corresponding 90% confidence interval for the mean does not exceed 5 m.u. is \(n \geq 173\).

1.4. The interval \([306.08\,;\,313.92]\) is a 95% confidence interval for the population mean, but it cannot have been constructed from the sample described above. (Comment.)

Exercise 2

Let \(X\) be the closing price (in euros) of shares “xyz” on the stock exchange. Assume that \(X\) follows a Normal distribution with a known standard deviation of \(\sigma = 1\) euro. A random sample of \(n = 16\) trading days was drawn, yielding an average closing price of \(\bar{x} = 9.5\) euros.

2.1. A point estimate for the true mean closing price of these shares is €9.50.

2.2. The estimator used in 2.1 to estimate the mean closing price is an unbiased estimator.

2.3. A 90% confidence interval for the true mean closing price of the shares, based on the sample collected, is:

2.4. The maximum margin of error for \(\mu\), at the same 90% confidence level, is 0.41125.

2.5. If the population variance were unknown, we would obtain a more precise confidence interval.

Exercise 3

Based on a random sample of \(n = 16\) observations drawn from a Normal population, the following confidence interval for the expected value was constructed using standard methodology:

\[[7.398\,;\,12.602]\]

3.1. From the sample, the corrected sample standard deviation was \(s' = 3.872\). The confidence level that can be assigned to the interval above is:

3.2. Based on the same sample, a 95% confidence interval for the population variance is:

3.3. Suppose the true variance is \(\sigma^2 = 36\). If a 95% confidence interval for the mean is to be constructed such that its width does not exceed 6.5, the minimum sample size is:

Exercise 4

The daily revenue of company XPTO is a random variable with an unknown distribution. A random sample of 91 days was collected, yielding a corrected sample standard deviation of \(s' = 385\) €.

4.1. With the information collected, all necessary conditions to determine a confidence interval for the true population variance (daily revenue of XPTO) are met.

Assuming all necessary conditions are met, the following confidence interval for the true population variance was constructed:

\[[112{,}922.8178\,;\,203{,}211.8756]\]

4.2(i). For the same confidence level, the corresponding confidence interval for the true population standard deviation is \([336.04\,;\,450.79]\).

4.2(ii). The confidence level of the interval above is 95%.

Exercise 5

In a certain district, 840 out of 2,000 voters surveyed in a poll declared they intend to vote for Party A.

5.1. A 95% confidence interval for the percentage of voters supporting Party A is:

5.2. If instead 4,000 voters had been surveyed and 1,680 had declared a preference for Party A, the 95% confidence interval would now be \([0.4047\,;\,0.4353]\).

5.3. Comparing the two sub-exercises above, we can say that increasing the sample size yields a confidence interval that is more .

Exercise 6

In a sample of 200 individuals who commute between home and work, 50 declared they would be willing to switch to the metro if it extended to their residential area.

6.1. A 95% confidence interval for the true proportion of individuals in these conditions is \([0.1899\,;\,0.3100]\).

6.2. The sample size required to halve the width of this interval, maintaining the same confidence level and the same estimate for the proportion, should be: