Statistics II – Hypothesis Testing: Exercises

This is a set of exercises prepared by the teaching faculty for Statistics II – Applied Statistics for Finance, covering Parametric Hypothesis Testing (tests for the mean, variance/standard deviation, and proportion), 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

A can manufacturer claims that the mean content of its cans is at least 250 grams. Assume that the filling process follows a Normal distribution with \(\sigma = 10\) g. To evaluate this claim, a random sample of \(n = 16\) cans was drawn, yielding a sample mean of \(\bar{x} = 245\) g.

1.i. The hypotheses that allow the manufacturer’s claim to be evaluated are:

\(H_0: \mu < 250 \text{ vs } H_1: \mu \geq 250\) \(H_0: \mu \leq 250 \text{ vs } H_1: \mu > 250\) \(H_0: \mu \geq 250 \text{ vs } H_1: \mu < 250\) \(H_0: \mu > 250 \text{ vs } H_1: \mu \leq 250\)

1.ii. At a significance level of 5%, we reject the hypothesis that the manufacturer is telling the truth.

1.iii. For \(\alpha > 0.0228\), the decision in 1.ii would change.

Exercise 2

A wine producer assures the regulatory authorities that his wine has a mean acidity level that does not exceed 0.5 g/l. The acidity level is assumed to follow a Normal distribution with unknown parameters. A sample of 20 bottles was observed, yielding a mean of 0.7 g/l and a corrected sample standard deviation of \(s' = 0.08\) g/l.

2. The statistical evidence leads us to conclude that the authorities should take action against the producer.

Exercise 3

Suppose \(n = 100\) and we wish to test whether the population mean equals 20 against the two-sided alternative that it does not equal 20. The sample mean is \(\bar{x} = 18\) and the corrected sample standard deviation is \(s' = 10\). The \(p\)-value for this test is:

0.0228 0.0456 0.5532 0

Exercise 4

The income (in m.u.) of residents of Tugoland is well modelled by a random variable \(X \sim N(\mu, \sigma)\). A random sample of \(n = 15\) residents was selected, yielding a sample mean of \(\bar{x} = 1.25\) m.u. and a corrected sample standard deviation of \(s' = 1.5\) m.u.

4.i. A trade union believes that the mean income of a resident does not exceed 1.1 m.u. To assess this claim, a hypothesis test was conducted with the following hypotheses:

\(H_0: \mu = 1.1 \text{ vs } H_1: \mu \neq 1.1\) \(H_0: \mu \leq 1.1 \text{ vs } H_1: \mu > 1.1\) \(H_0: \mu < 1.1 \text{ vs } H_1: \mu \geq 1.1\) \(H_0: \mu \geq 1.1 \text{ vs } H_1: \mu < 1.1\)

4.ii. For the test \(H_0: \mu = 1.1 \text{ vs } H_1: \mu \neq 1.1\), the observed value of the test statistic is 0.387.

4.iii. Consider the test \(H_0: \mu \leq 1.1 \text{ vs } H_1: \mu > 1.1\) with the decision rule that rejects the null hypothesis when \(\bar{x} > 1.782\). The significance level of the test for this decision rule is 0.04.

Exercise 5

The daily price (€) per litre of crude oil on the stock exchange is a random variable \(X \sim N(\mu, \sigma)\). To better understand this price, 30 days were randomly selected and the corresponding daily crude oil prices per litre were recorded. From the collected data, \(\bar{x} = 1.16\) € and \({s'}^{2} = 0.1567\) €² were obtained.

5. In a hypothesis test for the true mean daily price per litre of crude oil (with \(\sigma\) known), the critical region obtained was \((1.645\,;\,+\infty)\) and a \(p\text{-value} = 0.03\). Therefore, we reject the null hypothesis at the significance level fixed.

Exercise 6

Suppose you wish to test the following hypotheses at a 5% significance level:

\[H_0: \mu = \mu_0 \text{ vs } H_1: \mu \neq \mu_0\]

6.a. The probability of a Type I error is 0.025.

6.b. The test statistic, under \(H_0\), follows a \(t(n-1)\) distribution, where \(n\) is the size of the random sample collected.

Exercise 7

A watch manufacturer wishes to assess the variability of its product. To this end, he recorded the deviations (in seconds) of 20 watches relative to a high-precision timepiece after one month, obtaining \(s' = 2.3\) s. Based on long experience with watches of this brand, the manufacturer believes that the true standard deviation is at most 2 seconds.

7. The sample data lead to the conclusion that the manufacturer’s assumption is correct. (Assume any population conditions you consider appropriate.)

Exercise 8

It is known that the unemployment rate in a given region is 9%. With the aim of implementing support measures for the unemployed, 500 workers were randomly selected from the region, of whom 50 were found to be unemployed.

8.i. For the test \(H_0: p = 0.09 \text{ vs } H_1: p \neq 0.09\): when \(H_0\) is true, \((\hat{p} - 0.09)\,/\,0.013\) is a statistic with an approximately \(N(0,1)\) distribution.

8.ii. For the test \(H_0: p \leq 0.09 \text{ vs } H_1: p > 0.09\), we reject \(H_0\) based on the \(p\)-value.

Exercise 9

In a parametric hypothesis test, the null hypothesis is rejected at a significance level of 5%. Then:

\(H_0\) is rejected at a significance level of 1%. \(H_0\) is not rejected at a significance level of 1%. \(H_0\) may or may not be rejected at a significance level of 1%. Since \(H_0\) is rejected at 5%, it makes no sense to test at 1%.

Exercise 10

Which of the following is not necessary to compute the \(p\)-value?

Knowledge of whether the test is one-tailed or two-tailed. The observed value of the test statistic. The significance level. All of the above alternatives.

Exercise 11

A Type II error occurs when:

The null hypothesis is incorrectly accepted when it is false. The null hypothesis is incorrectly rejected when it is true. The sample mean differs from the population mean. The test is biased.