8.1: What is a hypothesis test?

Hypothesis tests

1. What is a hypothesis test?

2. Elementos of a test: null and alternative hypotheses, types of error, significance level, critical region

3. Tests for the mean of a normal population 4. Tests for a proportion

5. Other tests

Recommended reading:

 Capítulos 22 y 23 del libro de Peña y Romo (1997)

8.1: What is a hypothesis test?

A hypothesis is an affirmation made about the population.

The hypothesis is parametric if it refers to the values taken by one of the population parameters.

For example: “the population mean is positive” (μ > 0).

A hypothesis test is a statistical technique for judging whether or not the data provide evidence to confirm or reject the hypothesis.

Example

Después de la decisión de salir de Kosovo, es natural pensar que la popularidad de la Ministra de Defensa haya bajado.

Se midieron las valoraciones de Carme de 10 estudiantes antes y después de la crisis y las diferencias son:

-2, -0.4, -0.7, -2, +0.4, -2.2, +1.3, -1.2, -1.1, -2.3

La mayoría de los datos son negativos, pero ¿proporcionan estos datos evidencia de que el nivel medio de popularidad de Carme ha bajado?

La media estimada a partir de los datos es x = -1.02.

¿Refleja esta estimación un auténtico descenso en el nivel medio de popularidad? ¿Se debe el resultado a razones puramente aleatorias?

8.2: Elements of a hypothesis test

The hypothesis for which we wish to find evidence is called the alternative or experimental hypothesis. This is denoted by H₁.

< 0

The contrary hypothesis to H₁ is called the null hypothesis. This is denoted by H₀.

= 0

As we want to see whether Carmes mean rating really has gone down, then we wish to test H₀ : μ = 0 versus H₁ : μ < 0

The basis reasoning for carrying out a hypothesis test is as follows:

1. Suppose that H₀ is true, μ= 0.

2. Is the result obtained from the data (x = -1.02) very unlikely given this hypothesis?

3. If it is unlikely, the data provide evidence against H₀ and in favour of H₁.

To carry out this analysis, we need to see which values we would expect x to take if H₀ were true.

To simplify the problem, suppose that the population is normally distributed with known variance = 1.

Recall that

If H₀ is true, we have

To see if the observed mean is compatible with = 0, calculate

And compare this value with the standard normal distribution.

As -3,2255 is a very improbable value for the N(0, 1) distribution, from the tables, we have that P(Z < -3.2255) < 0.001), the data provide strong

evidence against H₀ and in favour of H₁.

Types of error

H

is true H

is true

Don’t reject H

Correct decision Type II error Reject H

Type I error Correct decision

Which of the two errors is more serious?

The significance level and the critical region We can control the type I error by fixing (a priori) the significance level, = P(reject H₀|H₀ is true)

Typical values are = 0,1 or 0,05 or 0,01.

Given the significance level, the critical or rejection region is the set of values of the test statistic such that H₀ is rejected.

Let = 0.05. Then H₀ is rejected if

that is if the sample mean is below -0.52. Poniendo = 0.025

The p-value

For small values of , it is more difficult to reject the null hypothesis. The minimum value of for which H₀ would be rejected given the data is called the p-value.

The p-value is interpreted as a measure of the statistical evidence in favour of H₁ (or against H₀): when the p-value is small, this represents strong evidence in favour of H₁.

Z_p = 3.2255 implies that p = 0.00063. There is lots of evidence against H₀ and in favour of H₁.

8.3: Tests for the mean of a normal population (known variance)

H₀ H₁ Rejection region

=

≤

=

≥

=

≠

Onesidedtests Twosidedtest

8.4: Tests for a proportion

H₀ H₁ Rejection region

p = p

p ≤ p

p = p

p ≥ p

p = p

p ≠ p

Onesidedtests Twosidedtest

Example

In the last elections, 40% of getafenses voted for the PSOE. In a study of 100 people, 37 said they would vote PSOE next time.

Calculate a 95% confidence interval for the probability that a getafense says they will vote PSOE in the next election.

Is there any evidence to suggest that this probability is below 0,4? Use a 5% significance level.

8.5: Other tests

1. Mean of a normal population (unknown variance) 2. Difference in the means of 2 normal populations

a) Known variances

b) Unknown but equal variances c) Unknown variances