Hypothesis Testing

Exercise 1

We want to study the energy efficiency of a chemical reaction that is documented having a nominal energy efficiency of \(90\%\). Based on previous experiments on the same reaction, we know that the energy efficiency is a Gaussian random variable with unknown mean \(\mu\) and variance equal to \(2\). In the last \(5\) days, the plant has given the following energy efficiencies (in percentage):

\[ 91.6, \quad 88.75, \quad 90.8, \quad 89.95, \quad 91.3 \]

Is the data in accordance with the specifications?
What is a point estimate of the mean energy efficiency?
Does that mean that the data significantly prove that the mean energy efficiency is larger than the expected nominal value?

Exercise 2

A study about air pollution done by a research station measured, on \(8\) different air samples, the following values of a polluant concentration (in \(\mu\)g/m\(^2\)):

\[ 2.2 \quad 1.8 \quad 3.1 \quad 2.0 \quad 2.4 \quad 2.0 \quad 2.1 \quad 1.2 \]

Assuming that the sampled population is normal,

Can we say that the mean polluant concentration is present with less than \(2.5 \mu\)g/m\(^2\)?
Can we say that the mean polluant concentration is present with less than \(2.4 \mu\)g/m\(^2\)?
Is the normality hypothesis essential to justify the method used?

Exercise 3

A medical inspection in an elementary school during a measles epidemic led to the examination of \(30\) children to assess whether they were affected. The results are in a tibble exam which contains the following:

# A tibble: 30 × 2
      Id Status 
   <int> <chr>  
 1     1 Healthy
 2     2 Healthy
 3     3 Healthy
 4     4 Healthy
 5     5 Healthy
 6     6 Healthy
 7     7 Healthy
 8     8 Healthy
 9     9 Healthy
10    10 Healthy
# ℹ 20 more rows

Let \(p\) be the probability that a child from the same school is sick.

Determine a point estimate \(\widehat{p}\) for \(p\).
The school will be closed if more than 5% of the children are sick. Can you conclude that, statistically, this is the case? Use a significance level of 5%.

Exercise 4

The capacities (in ampere-hours) of \(10\) batteries were recorded as follows:

\[ 140, \quad 136, \quad 150, \quad 144, \quad 148, \quad 152, \quad 138, \quad 141, \quad 143, \quad 151 \]

Estimate the population variance \(\sigma^2\).
Can we claim that the mean capacity of a battery is greater than 142 ampere-hours ?
Can we claim that the mean capacity of a battery is greater than 140 ampere-hours ?
Can we claim that the standard deviation of the capacity is less than 6 ampere-hours ?

Exercise 5

A company produces barbed wire in skeins of \(100\)m each, nominally. The real length of the skeins is a random variable \(X\) distributed as a \(\mathcal{N}(\mu, 4)\). Measuring \(10\) skeins, we get the following lengths:

\[ 98.683, 96.599, 99.617, 102.544, 100.110, 102.000, 98.394, 100.324, 98.743, 103.247 \]

Perform a conformity test at significance level \(\alpha = 5\%\).
Determine, on the basis of the observed values, the p-value of the test.

Exercise 6

In an atmospheric study the researchers registered, over \(8\) different samples of air, the following concentration of COG (in micrograms over cubic meter):

\[ 2.3;\; 1.7;\; 3.2;\; 2.1;\; 2.3;\; 2.0;\; 2.2;\; 1.2 \]

Using unbiased estimators, determine a point estimate of the mean and variance of COG concentration.

Assume now that the COG concentration is normally distributed.

Using a suitable statistical tool, establish whether the measured data allow to say that the mean concentration of COG is greater than \(1.8\) \(\mu\)g/m\(^3\).

Exercise 7

On a total of \(2350\) interviewed citizens, \(1890\) approve the construction of a new movie theater.

Perform an hypothesis test of level \(5\%\), with null hypothesis that the percentage of citizens that approve the construction is at least \(81\%\), versus the alternative hypothesis that the percentage is less than \(81\%\).
Compute the \(p\)-value of the test.
[difficult] Determine the minimum sample size such that the power of the test with significance level \(\alpha = 0.05\) when the real proportion \(p\) is \(0.8\) is at least \(50\%\).

Exercise 8

A computer chip manufacturer claims that no more than \(1\%\) of the chips it sends out are defective. An electronics company, impressed with this claim, has purchased a large quantity of such chips. To determine if the manufacturer’s claim can be taken literally, the company has decided to test a sample of \(300\) of these chips. If \(5\) of these \(300\) chips are found to be defective, should the manufacturer’s claim be rejected?

Exercise 9

To determine the impurity level in alloys of steel, two different tests can be used. \(8\) specimens are tested, with both procedures, and the results are written in the following table:

specimen n.	1	2	3	4	5	6	7	8
Test 1	1.2	1.3	1.7	1.8	1.5	1.4	1.4	1.3
Test 2	1.4	1.7	2.0	2.1	1.5	1.3	1.7	1.6

Assume that the data are normal.

based on the data in the table, can we state that at significance level \(\alpha=5\%\) the Test 1 and 2 give a different average level of impurity?
based on the data in the table, can we state that at significance level \(\alpha=1\%\) the Test 2 gives an average level of impurity greater than Test 1?

Exercise 10

A sample of \(300\) voters from region A and \(200\) voters from region B showed that the \(56\%\) and the \(48\%\), respectively, prefer a certain candidate. Can we say that at a significance level of \(5\%\) there is a difference between the two regions?

Exercise 11

In a sample of \(100\) measures of the boiling temperature of a certain liquid, we obtain a sample mean \(\overline{x} = 100^{o}C\) with a sample variance \(s^2 = 0.0098^{o}C^2\). Assuming that the observation comes from a normal population:

What is the smallest level of significance that would lead to reject the null hypothesis that the variance is \(\leq 0.015\)?
On the basis of the previous answer, what decision do we take if we fix the level of the test equal to \(0.01\)?