Solved Examples on Type I and Type II Errors

Samuel Dominic Chukwuemeka (SamDom For Peace)

Unless specified otherwise: identify Type I and Type II errors.
The names of the towns (in italics) are written to make you smile.
Yes, those towns exist. 😊
I want to make this topic fun as much as I can.

(1.) The proportion of people in the town of Scrabble, West Virginia whow write with their left hand is 0.15

Type I error:
Reject the claim that the proportion of people in the town who write with their left hand is $0.15$, when that proportion is actually true.

Type II error:
Fail to reject the claim that the proportion of people in the town who write with their left hand is $0.15$, when that proportion is actually different from $0.15$

(2.) Three years ago, the mean price of a single-family home in the village of Loving, New Mexico was $\$100,000$
A realtor believes that the mean price has increased since then.

Type I error:
Reject the hypothesis that the mean price of a family home in the village was $\$100,000$, when it was actually the true average price.

Type II error:
Do not reject the hypothesis that the average price of a family home in the village was $\$100,000$ when the actual average price was greater than $\$100,000$

(3.) A bottle contains a label stating that it contains Spring Valley pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains Merck pills with 40 mg of lisinopril that is used to treat high blood pressure.
Identify which one of the following errors is most serious, explain why it is most serious, and characterize the error as being a type I error or a type II error.

(a.) For lisinopril, fail to reject H₀: μ = 40mg when the mean is actually different from 40mg.

(b.) For lisinopril, reject H₀: μ = 40mg when the mean is actually equal to 40mg.

(c.) For vitamin C, fail to reject H₀: μ = 500mg when the mean is actually different from 500mg.

(d.) For vitamin C, reject H₀: μ = 500mg when the mean is actually equal to 500mg.

What are the dangers of consuming too much or too little vitamin C as compared to the dangers of consuming too much or too little blood pressure medication?
Consuming too much or too little of blood pressue medication is more serious than consuming too much or too little Vitamic C.
So, options (c.) and (d.) are eliminated.
So, let s focus on options (a.) and (b.)

A type I error is the mistake of rejecting the null hypothesis when it is actually true.
This would mean: not consuming the Merck pills with 40mg of lisinopril when in fact the pills have 40mg of lisinopril

A type II error is the mistake of failing to reject the null hypothesis when it is actually false.
This would mean: consuming the Merck pills with either more than or less than 40mg of lisinopril.

The more serious error in this case is the type II error because comsuming more than 40mg of lisinopril is overdose of the blood pressure medication while consuming less than 40mg of lisinopril is underdose of the blood pressure medication.

Error (a.) is most serious, since this error is the most likely to result in a person consuming a potentially dangerous quantity of a substance.
This error would be a type II error.

(4.) Suppose the consequences of making a Type I error are severe, which of these levels of significance would you choose: 1%? 5%, or 10%?

Use the 1% significance level because it is the smallest that was given.
Recall that the level of significance is the probability of making a Type I error.
You really want it to be very small.

(5.) A bottle contains a label stating that it contains pills with 500 mg of vitamin C.
Another bottle contains a label stating that it contains pills with 325 mg of aspirin.
When testing claims about the mean contents of the pills, which of these would have more serious implications:
(a.) The rejection of the vitamin C claim OR the rejection of the aspirin claim?
(b.) Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?

(a.) The rejection of the claim about the aspirin is more serious because the wrong aspirin dosage could cause more serious adverse reactions than a wrong vitamin C dosage.
(b.) It would be wise to use a smaller significance level for testing the claim about the aspirin.

(6.) A researcher is testing someone who claims to have ESP (ExtraSensory Perception) by having that person predict whether a coin will come up heads or tails.
The null hypothesis is that the person is guessing and does not have ESP, and the population proportion of success is 0.50.
The researcher tests the claim with a hypothesis test, using a significance level of 0.05.
The probability of concluding that the person has ESP when in fact she or he ................... have ESP is ..........

The level of significance is the probability of making a Type I error.
This means that the level of significance is the probability of rejecting the null hypotheses when it is true.
The level of significance given in the question is 0.05
Therefore, the probability of concluding that the person has ESP when in fact she or he does not have ESP is 0.05

(9.) A teacher is giving an exam with 20 multiple-choice questions, each with four possible answers.
The teacher's null hypothesis is that a certain student is just guessing, and the population proportion of success is 0.25.
Suppose the teacher conducts a test with a significance level of 0.01.
Write a sentence describing the significance level in the context of the hypothesis test.

A. The probability that the student is guessing is 0.01.

B. The probability of saying the student is guessing when the student is actually not guessing is 0.01.

C. The probability of saying the student is not guessing when the student is actually guessing is 0.01.

D. The probability that the student is not guessing is 0.01.

E. The probability of saying the student is guessing when the student is guessing is 0.01.

F. The probability of saying the student is not guessing when the student is not guessing is 0.01.

The significance level is the probability of making a Type I error.
This means that the significance level is the probability of rejecting the null hypotheses when it is true.
In the context of the question, this means that:
The probability of saying the student is not guessing when the student is actually guessing is 0.01.

(11.) An alternative hypothesis is: p < 0.65
To test this hypothesis, a random sample of size 360 is taken.
Calculate the probability of making a Type II error if the alternative proportion is 60%.
Use a 5% significance level.

$ H_0:\;\; p = 0.65 \\[3ex] H_1;\;\; p \lt 0.65 \\[5ex] n = 360 \\[3ex] p = 0.65 \\[3ex] q = 1 - p \\[3ex] q = 1 - 0.65 \\[3ex] q = 0.35 \\[3ex] z_{\dfrac{\alpha}{2}} = -1.645 \\[7ex] \hat{p_c} = z_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{pq}{n}} + p \\[7ex] \hat{p_c} = -1.645 * \sqrt{\dfrac{0.65 * 0.35}{360}} + 0.65 \\[5ex] \hat{p_c} = 0.6086471589 \\[3ex] p_a = 60\% = 0.6 \\[3ex] q_a = 1 - p_a \\[3ex] q_a = 1 - 0.6 \\[3ex] q_a = 0.4 \\[3ex] z = \dfrac{\hat{p_c} - p_a}{\sqrt{\dfrac{pq}{n}}} \\[8ex] z = \dfrac{0.6086471589 - 0.6}{\sqrt{\dfrac{0.65 * 0.35}{360}}} \\[8ex] z = 0.3439806315 \\[3ex] z \approx 0.34 \\[3ex] \beta = P(z \lt -0.34) = 0.36693 $

(12.) In a clinical trial of a drug intended to help the residents of the town of Experiment, Georgia; 133 subjects were treated with the drug for 11 weeks, and 15 subjects experienced abdominal pain.
Someone claimed that more than 0.08 of the drug’s users experienced abdominal pain.
The claim was supported with a hypothesis test conducted with a 5% significance level.
Using 0.17 as an alternative value of P, the power of the test is 0.96.
Interpret the power of the test.

The power of 0.96 shows that there is a 96% chance of rejecting the null hypothesis of P = 0.08 when the true proportion is 0.17.
This means that if the proportion of users who experienced abdominal pain was 0.17, then there is a 96% chance of supporting the claim that the proportion of users who experienced abdominal pain was greater than 0.08.

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