MATH 1680

If the consequences of making a Type I error are​ severe, would you choose the level of​ significance, α​, to equal​ 0.01, 0.05, or​ 0.10?

0.01

List the 3 steps in hypothesis testing.

1. Make a statement regarding the nature of the population. 2. Collect evidence (sample data) to test the statement 3. Analyze the data to assess the plausibility of the statement

Give the definition of a P-value.

A P-value is the probability of observing a sample statistic as extreme as or more extreme than one observed under the assumption that the statement in the null hypothesis is true. Stated another way, the P-value is the likelihood or probability that a sample will result in a statistic such as the one obtained if the null hypothesis is true.

State the definition of hypothesis testing.

A procedure based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

What is a hypothesis?

A statement regarding a characteristic of one or more populations.

State the definition of the null hypothesis.

A statement to be tested. The null hypothesis is a statement of no change, no effect, or no difference and is assumed true until evidence indicates otherwise.

Why do we test statements about a population parameter using sample data?

Because it is usually impossible or impractical to gain access to the entire population.

In a jury trial, what decision is equivalent to making a Type II error?

Declaring a guilty person "not guilty"

In a jury trial, what decision is equivalent to making a Type I error?

Declaring an innocent person guilty

What type of error is called a Type II error?

Do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a Type II error.

What jury decision is associated with rejecting the null hypothesis?

Guilty

Explain how to determine whether the null hypothesis should be rejected using the P-value approach.

If the probability of getting a sample statistic as extreme as or more extreme than the one obtained is small under the assumption that the statement in the null hypothesis is true, reject the null hypothesis.

For the sampling distribution of p^ to be approximately normal, we require that np(1-p) be at least 10.

If this requirement is not satisfied we use the binomial probability formula to determine the P-value.

What type of tests are referred to as one-tailed tests?

Left and right tailed tests

What is at the​ "heart" of hypothesis testing in​ statistics?

Make an assumption about​ reality, and collect sample evidence to determine whether it contradicts the assumption.

Is the null hypothesis ever declared "true"?

No, it is either rejected or not rejected

What jury decision is associated with failing to reject the null hypothesis?

Not guilty

In a jury trial, what are the null and alternative hypotheses?

Null hypothesis: innocent Alternative hypothesis: guilty

Why is the level of significance not always set at α=0.01

Reducing the probability of making a Type I error increases the probability of making a Type II error, β. Using our court analogy from the video explaining Figure 1, a jury is instructed that the prosecution must provide proof of guilt "beyond all reasonable doubt." This implies that we are choosing to make α small so that the probability of convicting an innocent person is very small. The consequence of the small α, however, is a large β, which means many guilty defendants will go free. For now, we are content to recognize the inverse relation between α and β. (As one goes up, the other goes down.

What type of error is called a Type I error?

Reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error.

It is important to recognize that we never accept the null hypothesis.

Sample evidence can never prove the null hypothesis to be true. By not rejecting the null hypothesis, we are saying that the evidence indicates that the null hypothesis could be true or that the sample evidence is consistent with the statement in the null hypothesis.

State the five steps for testing a hypothesis about a population proportion, p.

Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways: Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error . Step 3 (By Hand) Step 3 (Using Technology): compute the test statistic Step 4: If P-value <α, reject the null hypothesis. Step 5: State the conclusion

What does the choice of the level of significance depend on?

The choice of the level of significance depends on the consequences of making a Type I error. If the consequences are severe, the level of significance should be small (say, α=0.01). However, if the consequences are not severe, a higher level of significance can be chosen (say, α=0.05 or α=0.10).

What does the level of significance represent?

The level of significance, α, is the probability of making a Type I error

What determines the structure of the alternative hypothesis (two-tailed, left-tailed, or right-tailed?)

The statement we are trying to gather evidence for.

List the three ways to set up the null and alternative hypotheses.

Two tailed test Equal versus not equal hypothesis H0 : parameter = some value H1 : parameter does not equal some value Left-tailed test 2. Equal versus less than H0 : parameter = some value H1 : parameter < some value Right-tailed test 3. Equal versus greater than H0 : parameter = some value H1 : parameter > some value

Give the definition of what it means for a result to be statistically significant.

When observed results are unlikely under the assumption that the null hypothesis is true, we say that the result is statistically significant and we reject the statement in the null hypothesis.

Explain how to make a decision about the null hypothesis when performing a two-tailed test using confidence intervals.

When testing H0: p=p0 versus H1: p≠p0, if a (1−α)⋅100% confidence interval contains p0, we do not reject the null hypothesis. However, if the confidence interval does not contain H0: p=p0 versus H1: p≠p0, if a (1−α)⋅100% confidence interval contains p0, we do not reject the null hypothesis. However, if the confidence interval does not contain p0, we conclude that p≠ p0 at the level of significance α.

When there are small sample​ sizes, the evidence against the statement in the null hypothesis must be __________ One should be wary of studies that _____________ the null hypothesis when the test was conducted with a small sample size.

substantial; do not reject

What are the three conditions that must be satisfied before testing a hypothesis regarding a population proportion, p?

the sample is obtained by simple random sampling or the data result from a randomized experiment ; np0(1−p0)≥10 where p0 is the proportion stated in the null hypothesis; and the sampled values are independent of each other. This means that the sample size is no more than 5% of the population size (n≤0.05N).

A criterion for testing hypotheses is to determine how likely the observed sample proportion is:

under the assumption that the statement in the null hypothesis is true.

What symbols do we use to denote the probability of making a Type I error and the probability of making a Type II error?

α = P(Type I error)=P(rejecting H0 when H0 is true) β=P(Type II error) = P(not rejecting H0 when H1 is true)