Question 1

If the null hypothesis is not rejected, there is strong statistical evidence that the null hypothesis is true.

Accepted Answer

If the null hypothesis is not rejected, it means that there is not enough evidence to reject it, but it doesn't necessarily mean that it is true. It is possible that a type II error occurred, where a false null hypothesis was not rejected. Therefore, there is not strong statistical evidence that the null hypothesis is true.

Question 2

Children as young as 2 years of age, upon seeing an object placed under a pillow in a familiar setting at home, will understand to look for it after an interval of time and be able to find it. Investigators believe this capability will be less pronounced in a laboratory situation, where the child is away from the familiar setting of home. Let π denote the proportion of 2-year-olds than have this understanding in the home situation, and suppose that π = .35. The investigators wish to determine whether the proportion that remembers is less when the child is away from home. 
a) What is the appropriate null hypothesis? 
b) What is the appropriate alternative hypothesis? 
c) In your own words, distinguish between a Type I and a Type II error in this context.

Accepted Answer

a) H₀ : π = .35 b) Ha : π < .35 c) A Type I error is to decide that children remember less when outside of the home environment, when in fact, they remember equally well. A Type II error is to decide that children remember as well in the laboratory setting as in the home setting, when in fact they remember less away from home.

Question 3

The level of significance of a test is the probability of making a type I error, given that the null hypothesis is true.

Accepted Answer

This is the definition of the level of significance or alpha. It represents the probability of rejecting a true null hypothesis, also known as a type I error.

Question 4

Small P-values indicate that the observed sample is inconsistent with the null hypothesis.

Accepted Answer

A small P-value means that the probability of obtaining the observed results under the assumption that the null hypothesis is true is very low. Therefore, it suggests that the null hypothesis is unlikely to be true and the observed sample is inconsistent with it.

Question 5

The power of a test is the probability of failing to reject the null hypothesis.

Accepted Answer

The power of a test is the probability of correctly rejecting the null hypothesis when it is false, not the probability of failing to reject it.

Question 6

Psychologists have noted that people tend make classifications based on their "typicality." For example, a dog is more readily classified as a mammal than is a whale, because dogs are "more typical" of mammals than are whales. Previous large-scale studies have established the following for songbirds such as robins: when young children are shown a picture of a songbird and asked if it is a bird they respond "yes" in an average of 750 milliseconds. Do children regard chickens as representative of birds? If so, are they more representative or less representative of birds than are songbirds? Let μ represent the mean time it takes children to respond to a question about whether a chicken is a bird. The investigators wish to determine whether the classification time differs from the time needed to classify songbirds. 
a) What is the appropriate null hypothesis? 
b) What is the appropriate alternative hypothesis? 
c) In your own words, distinguish between a Type I and a Type II error in this context.

Accepted Answer

The answer of Psychologists have noted that people tend make...

Question 7

The statement s² = 100 is a statistical hypothesis.

Accepted Answer

The statement "s² = 100" is not a statement that can be interpreted as a statistical hypothesis. It is unclear what data is being referred to and what is being tested. Therefore, the answer is false.

Question 8

In your own words, distinguish between "statistical" and "practical" significance.

Accepted Answer

The answer of In your own words, distinguish between "statistical"...

Question 9

The choice of the alternative hypothesis depends on the objectives of the study.

Accepted Answer

The alternative hypothesis should reflect the researcher's research question, and the objectives of the study determine the research question. Therefore, the choice of the alternative hypothesis depends on the objectives of the study.

Question 10

Explain in your own words what a hypothesis test is.

Accepted Answer

The answer of Explain in your own words what a...

Question 11

In cities and towns on the borders between states there is a significant amount of flight across state lines to avoid high state taxes on gasoline. Some states (such as Iowa) have large rivers for borders and tolls to cross the bridges. Do these tolls impede traffic to other states to get cheaper gasoline? To test this hypothesis, an experimental Toll-Free Week will be instituted at the Farmington Bridge, where currently 50 cars per day drive out of the state. Let μ denote the true average number of border crossings per day at Farmington if there were no toll. 
a) What is the appropriate null hypothesis? 
b) What is the appropriate alternative hypothesis? 
c) In your own words, distinguish between a Type I and a Type II error in this context.

Accepted Answer

The answer of In cities and towns on the borders...

Question 12

Explain in your own words the distinction between a null and alternative hypothesis.

Accepted Answer

The answer of Explain in your own words the distinction...

Question 13

The Environmental Protection Agency (EPA) is charged with monitoring the environment. One aspect of this is keeping track of "acid rain," a broad term describing the fall of water through an acidic atmosphere. Acidity is measured on the pH scale, where pure water has a pH of 7.0. Normal rain is slightly acidic because carbon dioxide dissolves into it, and thus has a pH of about 5.5. (A lower pH indicates greater acidity.) Suppose the EPA wishes to determine whether a particular area is subject to acid rain. Let μ denote the true average for pH in this area. 
a) What is the appropriate null hypothesis? 
b) What is the appropriate alternative hypothesis? 
c) In your own words, distinguish between a Type I and a Type II error in this context.

Accepted Answer

The answer of The Environmental Protection Agency (EPA) is charged...

Question 14

It is customary to say that the result of a hypothesis test is statistically significant when the P-value is smaller than α.

Accepted Answer

This statement is true. If the P-value is smaller than the chosen significance level (α), then we reject the null hypothesis and consider the result to be statistically significant.

Question 15

β is called the observed significance level.

Accepted Answer

β is the probability of committing a Type II error, whereas the observed significance level is calculated based on the data and represents the probability of observing a test statistic as extreme as or more extreme than the one observed, assuming the null hypothesis is true.

Question 16

Suppose that a study has been published linking arsenic, which is commonly found in tap water, to increased cancer rates. The study also stated that arsenic levels below 10 ppm could be considered harmless. Suppose that residents of Hacienda Heights, CA, are worried about the arsenic level in the local tap water. 
a) What hypothesis should they test? 
b) Describe a Type I and Type II error in this context.

Accepted Answer

The answer of Suppose that a study has been published...

Question 17

All other things being equal, choosing a smaller value of α will increase the probability of making a type II error.

Accepted Answer

This is because by choosing a smaller value of α, we are being more conservative with our hypothesis testing and increasing the likelihood that we will fail to reject the null hypothesis even if it is false (i.e. making a type II error).

Question 18

For tests of hypotheses about μ, β decreases as the sample size increases if the level of significance stays the same.

Accepted Answer

As the sample size increases, the standard error decreases, leading to a narrower distribution of sample means. This results in a larger t-value or z-value for a given level of significance and hence decreases the likelihood of committing a Type II error. Therefore, the power of the test increases and the probability of a Type II error (β) decreases.

Question 19

A type II error is made by failing to reject a false null hypothesis.

Accepted Answer

This statement accurately describes a type II error, which occurs when we do not reject a null hypothesis that is actually false. This can lead to accepting an incorrect conclusion or missing a true effect or relationship.

Question 20

When performing tests of hypotheses, there are assumptions that must be met in order for the test to be appropriate. For the test of a hypothesis about a population mean, and given the raw data, describe how you would check the assumptions.

Accepted Answer

The answer of When performing tests of hypotheses, there are...

Quiz 10: Hypothesis Testing Using a Single Sample