Hypothesis Testing

Site:	Saylor Academy
Course:	CS250: Python for Data Science
Book:	Hypothesis Testing

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Date:	Saturday, May 18, 2024, 2:57 AM

Description

In addition to calculating confidence intervals, hypothesis testing is another way to make statistical inferences. This process involves considering two opposing hypotheses regarding a given data set (referred to as the null hypothesis and the alternative hypothesis). Hypothesis testing determines whether the null hypothesis can be accepted or rejected.

Null and Alternative Hypotheses
Outcomes and the Type I and Type II Errors
Distribution Needed for Hypothesis Testing

Null and Alternative Hypotheses

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.

H₀, the - null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H_a - , the alternative hypothesis: a claim about the population that is contradictory to H₀ and what we conclude when we reject H₀.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H₀ if the sample information favors the alternative hypothesis or do not reject H₀ or decline to reject H₀ if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H₀ and H_a:

H₀	H_a
equal (=)	not equal (≠) or greater than (>) or less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

Table 9.1

Note

H₀ always has a symbol with an equal in it. H_a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H₀ : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H_a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

Try It 9.1

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following:
H₀ : μ = 2.0

H_a : μ ≠ 2.0

Try It 9.2

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H₀ : μ __ 66
H_a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following:
H₀ : μ ≥ 5

H_a : μ < 5

Try It 9.3

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H₀ : μ __ 45
H_a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses.
H₀ : p ≤ 0.066

H_a : p > 0.066

Try It 9.4

On a state driver's test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H₀ : p __ 0.40
H_a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

Source: OpenStax, https://openstax.org/books/statistics/pages/9-introduction
This work is licensed under a Creative Commons Attribution 4.0 License.

Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth, or falseness, of the null hypothesis H₀ and the decision to reject or not. The outcomes are summarized in the following table:

ACTION	H₀ IS ACTUALLY	...
	True	False
Do not reject H₀	Correct outcome	Type II error
Reject H₀	Type I error	Correct outcome

Table 9.2

The four possible outcomes in the table are as follows:

The decision is not to reject H₀ when H₀ is true (correct decision).
The decision is to reject H₀ when, in fact, H₀ is true (incorrect decision known as a Type I error).
The decision is not to reject H₀ when, in fact, H₀ is false (incorrect decision known as a Type II error).
The decision is to reject H₀ when H₀ is false (correct decision whose probability is called the Power of the Test).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

$α$ = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

$β$ = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

α and β should be as small as possible because they are probabilities of errors. They are rarely zero.

The Power of the Test is 1 – β. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.

The following are examples of Type I and Type II errors.

Example 9.5

Suppose the null hypothesis, H₀, is: Frank's rock climbing equipment is safe.

Type I error: Frank does not go rock climbing because he considers that the equipment is not safe, when in fact, the equipment is really safe. Frank is making the mistake of rejecting the null hypothesis, when the equipment is actually safe!

Type II error: Frank goes climbing, thinking that his equipment is safe, but this is a mistake, and he painfully realizes that his equipment is not as safe as it should have been. Frank assumed that the null hypothesis was true, when it was not.

α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it).

Try It 9.5

Suppose the null hypothesis, H₀, is: the blood cultures contain no traces of pathogen X. State the Type I and Type II errors.

Example 9.6

Suppose the null hypothesis, H₀, is: a tomato plant is alive when a class visits the school garden.

Type I error: The null hypothesis claims that the tomato plant is alive, and it is true, but the students make the mistake of thinking that the plant is already dead.

Type II error: The tomato plant is already dead (the null hypothesis is false), but the students do not notice it, and believe that the tomato plant is alive.

α = probability that the class thinks the tomato plant is dead when, in fact, it is alive = P(Type I error). β = probability that the class thinks the tomato plant is alive when, in fact, it is dead = P(Type II error).

The error with the greater consequence is the Type I error. (If the class thinks the plant is dead, they will not water it).

Try It 9.6

Suppose the null hypothesis, H₀, is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

Example 9.7

It's a Boy Genetic Labs, a genetics company, claims to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H₀, is: It's a Boy Genetic Labs has no effect on gender outcome.

Type I error: This error results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It's a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, α.

Type II error: This error results when we fail to reject a false null hypothesis. In context, we would state that It's a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, β.

The error with the greater consequence would be the Type I error since couples would use the It's a Boy Genetic Labs product in hopes of increasing the chances of having a boy.

Try It 9.7

Red tide is a bloom of poison-producing algae - a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries montors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kilogram of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.

Example 9.8

A certain experimental drug claims a cure rate of at least 75 percent for males with a disease. Describe both the Type I and Type II errors in context. Which error is the more serious?

Type I: A patient believes the cure rate for the drug is less than 75 percent when it actually is at least 75 percent.

Type II: A patient believes the experimental drug has at least a 75 percent cure rate when it has a cure rate that is less than 75 percent.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75 percent of the time, this most likely will influence the patient's (and doctor's) choice about whether to use the drug as a treatment option.

Try It 9.8

Determine both Type I and Type II errors for the following scenario:

Assume a null hypothesis, H₀, that states the percentage of adults with jobs is at least 88 percent.

Identify the Type I and Type II errors from these four possible choices.

Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when that percentage is actually less than 88 percent
Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when the percentage is actually at least 88 percent
Reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when the percentage is actually at least 88 percent
Reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when that percentage is actually less than 88 percent

Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's t-distribution. (Remember, use a Student's t-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal). We perform tests of a population proportion using a normal distribution (usually $n$ is large).

Assumptions

When you perform a hypothesis test of a single population mean μ using a Student's t-distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed.

When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion $p$ , you take a simple random sample from the population. You must meet the conditions for a binomial distribution, which are the following: there are a certain number $n$ of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success $p$ . The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities $np$ and $nq$ must both be greater than five $(np > 5 \text{ and }nq > 5)$ . Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with $μ = p$ and $σ=\sqrt{\dfrac{pq}{n}}$ . Remember that $q = 1 – p$ .