Hypothesis Testing with One Sample

# Outcomes and the Type I and Type II Errors

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When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis *H _{0}* and the decision to reject or not. The outcomes are summarized in the following table:

ACTION | H IS ACTUALLY_{0} |
… |
---|---|---|

True | False | |

Do not reject H_{0} |
Correct Outcome | Type II error |

Reject H_{0} |
Type I Error | Correct Outcome |

The four possible outcomes in the table are:

**not to reject**when

*H*_{0}

*H*is true (correct decision)._{0}**reject**when

*H*_{0}**(incorrect decision known as aType I error).**

*H*is true_{0}**not to reject**when, in fact,

*H*_{0}**(incorrect decision known as a Type II error).**

*H*is false_{0}**reject**when

*H*_{0}**(**

*H*is false_{0}**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.

Suppose the null hypothesis, *H _{0}*, is: Frank’s rock climbing equipment is safe.

**Type I error**: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. **Type II error**: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.

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

**that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.**

*β*= probabilityNotice 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.)

Suppose the null hypothesis, *H _{0}*, is: the blood cultures contain no traces of pathogen

*X*. State the Type I and Type II errors.

Type I error: The researcher thinks the blood cultures do contain traces of pathogen *X*, when in fact, they do not.

Type II error: The researcher thinks the blood cultures do not contain traces of pathogen *X*, when in fact, they do.

Suppose the null hypothesis, *H _{0}*, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.

**Type I error**: The emergency crew thinks that the victim is dead when, in fact, the victim is alive. **Type II error**: The emergency crew does not know if the victim is alive when, in fact, the victim is dead.

** α = probability** that the emergency crew thinks the victim is dead when, in fact, he is really alive =

*P*(Type I error).

**that the emergency crew does not know if the victim is alive when, in fact, the victim is dead =**

*β*= probability*P*(Type II error).

The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)

Suppose the null hypothesis, *H _{0}*, is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

The error with the greater consequence is the Type II error: the patient will be thought well when, in fact, he is sick, so he will not get treatment.

It’s a Boy Genetic Labs claim 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, *H _{0}*, is: It’s a Boy Genetic Labs has no effect on gender outcome.

**Type I error**: This 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 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 of 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.

“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 (DMF) monitors 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 kg 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.

In this scenario, an appropriate null hypothesis would be*H _{0}*: the mean level of toxins is at most 800

*μ*g,

*H*:

_{0}*μ*

_{0}≤ 800

*μ*g.

**Type I error**: The DMF believes that toxin levels are still too high when, in fact, toxin levels are at most 800 *μ*g. The DMF continues the harvesting ban.

**Type II error**: The DMF believes that toxin levels are within acceptable levels (are at least 800 *μ*g) when, in fact, toxin levels are still too high (more than 800 *μ*g). The DMF lifts the harvesting ban. This error could be the most serious. If the ban is lifted and clams are still toxic, consumers could possibly eat tainted food.

In summary, the more dangerous error would be to commit a Type II error, because this error involves the availability of tainted clams for consumption.

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

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

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

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% 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.

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

Assume a null hypothesis, *H _{0}*, that states the percentage of adults with jobs is at least 88%.

Identify the Type I and Type II errors from these four statements.

Type I error: c

Type I error: b

# Chapter Review

In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A **Type I** error occurs when a true null hypothesis is rejected. A **Type II error** occurs when a false null hypothesis is not rejected.

The probabilities of these errors are denoted by the Greek letters *α* and *β*, for a Type I and a Type II error respectively. The power of the test, 1 – *β*, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.

# Formula Review

*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.

The mean price of mid-sized cars in a region is 32,000, but we conclude that it is not 32,000, but we conclude that it is 100,000 per year.

Twenty-nine percent of high school seniors get drunk each month.

Fewer than 5% of adults ride the bus to work in Los Angeles.

The mean number of cars a person owns in his or her lifetime is not more than ten.

About half of Americans prefer to live away from cities, given the choice.

Europeans have a mean paid vacation each year of six weeks.

The chance of developing breast cancer is under 11% for women.

Private universities mean tuition cost is more than 100,000, when it really is at least 100,000 when, in fact, it is less than 20,000, though in reality it is at most 20,000 when, in fact, it is more than $20,000.

For statements a-j in Exercise 9.109, answer the following in complete sentences.

- State a consequence of committing a Type I error.
- State a consequence of committing a Type II error.

When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. Suppose the null hypothesis is “the drug is unsafe.” What is the Type II Error?

- To conclude the drug is safe when in, fact, it is unsafe.
- Not to conclude the drug is safe when, in fact, it is safe.
- To conclude the drug is safe when, in fact, it is safe.
- Not to conclude the drug is unsafe when, in fact, it is unsafe.

b

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. The Type I error is to conclude that the percent of EVC students who attended is ________.

- at least 20%, when in fact, it is less than 20%.
- 20%, when in fact, it is 20%.
- less than 20%, when in fact, it is at least 20%.
- less than 20%, when in fact, it is less than 20%.

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?

The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours

- is more than seven hours.
- is at most seven hours.
- is at least seven hours.
- is less than seven hours.

d

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test, the Type I error is:

- to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher
- to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same
- to conclude that the mean hours per week currently is 4.5, when in fact, it is higher
- to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher

## Glossary

- Type 1 Error
- The decision is to reject the null hypothesis when, in fact, the null hypothesis is true.

- Type 2 Error
- The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.