Question 1

A researcher conducts a hypothesis test at the

\alpha = 0.05

significance level and calculates a p-value of 0.024. Which of the following is the most appropriate conclusion to draw regarding the null hypothesis (

H_0

)?

Accepted Answer

Reject

\alpha = 0.05

; there is convincing evidence for the alternative hypothesis because the p-value is less than the significance level.

Answer

Reject

\alpha = 0.05

; the results are not statistically significant because the p-value is greater than zero.

Answer

Accept

\alpha = 0.05

; the p-value is small enough to prove that the null hypothesis is the most likely explanation.

Answer

Fail to reject

\alpha = 0.05

; the p-value indicates that there is a 2.4% probability that the null hypothesis is true.

Question 2

An agricultural scientist is testing a new fertilizer and calculates a 95% confidence interval for the mean increase in crop yield to be

\alpha = 0.05

kilograms per plot. If the null hypothesis states that the mean increase is

H_0

, what can be concluded about the statistical significance of these results at the

\alpha = 0.05

level?

Accepted Answer

The results are statistically significant because the null value of 0 is not contained within the 95% confidence interval.

Answer

The results are not statistically significant because a confidence interval cannot be used to determine a p-value.

Answer

The results are statistically significant because the width of the interval is exactly 3.6.

Answer

The results are not statistically significant because the confidence interval does not include the value 5.

Question 3

In a clinical trial, a researcher finds a statistically significant difference in recovery times using a new medication with a p-value of

\alpha = 0.05

at a significance level of

H_0

. However, the mean recovery time only decreased by 2 hours in a 10-day recovery period. Which of the following statements correctly distinguishes between statistical significance and practical significance?

Accepted Answer

The results are statistically significant because

\alpha = 0.05

, but they may lack practical significance because a 2-hour reduction is negligible in a 10-day period.

Answer

The results have practical significance because any p-value below 0.05 automatically proves the treatment is highly effective for patients.

Answer

Statistically significant results always imply practical significance regardless of the actual magnitude of the observed difference.

Answer

The results are not statistically significant because the effect size of 2 hours is too small to be meaningful in a real-world setting.

Question 4

Suppose a researcher performs a hypothesis test and decides to lower the significance level from

\alpha = 0.05

to

H_0

. If the original results were statistically significant at the

\alpha = 0.05

level with a p-value of

0.03

, what happens to the conclusion under the new significance level?

Accepted Answer

The results are no longer statistically significant because the p-value of

\alpha = 0.05

is now greater than the new threshold of

H_0

.

Answer

The results remain statistically significant because the p-value is unaffected by the choice of

\alpha = 0.05

.

Answer

The null hypothesis is now more likely to be false because the standard for evidence has become more rigorous.

Answer

The p-value will automatically decrease to

\alpha = 0.05

to maintain the statistical significance of the study.

Question 5

A medical study reports a p-value of

\alpha = 0.05

for the effectiveness of a new drug compared to a placebo. Which of the following is the most accurate interpretation of this p-value in the context of statistical significance?

Accepted Answer

If the drug is actually no more effective than the placebo, the probability of observing an effect as large or larger than the one found in this study is 0.1%.

Answer

Because the p-value is so small, the drug is guaranteed to have a large and meaningful practical effect on all patients.

Answer

There is a 99.9% probability that the new drug is effective and will work for all patients.

Answer

The probability that the null hypothesis is true is exactly 0.001.

Question 6

A quality control manager at a lightbulb factory conducts a hypothesis test to determine if a new manufacturing process changes the mean life of the bulbs. The manager calculates a test statistic that results in a p-value of

\alpha = 0.05

. Using a standard significance level of

H_0

, which of the following is the most appropriate conclusion?

Accepted Answer

The manager should fail to reject the null hypothesis because the p-value is greater than the significance level, meaning there is not enough evidence to claim the mean life has changed.

Answer

The manager should accept the null hypothesis as true because the p-value is greater than

\alpha = 0.05

, proving the mean life is exactly the same as before.

Answer

The manager should reject the null hypothesis because a p-value of

\alpha = 0.05

shows that there is an 8% chance the new process is better.

Answer

The manager should reject the null hypothesis because the results are statistically significant at the 10% level, and 10% is close enough to 5%.

Question 7

An environmental scientist is testing whether a specific power plant is exceeding its permitted water temperature discharge. The null hypothesis is

\alpha = 0.05

°C and the alternative is

H_0

°C. The scientist calculates a test statistic that leads to a p-value of

\alpha = 0.05

. If the scientist uses a significance level of

0.03

, what is the correct evaluation of the statistical significance?

Accepted Answer

The results are not statistically significant because

\alpha = 0.05

, so we fail to reject the null hypothesis.

Answer

The results are statistically significant because

\alpha = 0.05

is a small probability, indicating the null hypothesis is likely false.

Answer

The results are statistically significant because the p-value is less than

\alpha = 0.05

, which is the default for all scientific studies.

Answer

The results are not statistically significant because the p-value must be exactly equal to the significance level to reject the null hypothesis.

Question 8

A researcher is testing a new weight-loss program with the null hypothesis

\alpha = 0.05

kg and the alternative hypothesis

H_0

kg. They calculate a test statistic that corresponds to a p-value of

\alpha = 0.05

. If the researcher uses a significance level of

0.03

, which of the following is the most accurate statistical conclusion?

Accepted Answer

Reject

\alpha = 0.05

; the results are statistically significant because the p-value is less than

H_0

.

Answer

Fail to reject

\alpha = 0.05

; there is a 3.8% probability that the weight-loss program is completely ineffective.

Answer

Reject

\alpha = 0.05

; the p-value indicates that there is only a 3.8% chance that the null hypothesis is false.

Answer

Accept

\alpha = 0.05

; the p-value is not small enough to prove that the weight-loss program works for every individual.

Question 9

A psychologist is studying the effectiveness of a new therapy. Using a significance level of

\alpha = 0.05

, the study yields a p-value of

H_0

. Which of the following statements is a correct evaluation of the statistical significance of this result?

Accepted Answer

The result is not statistically significant because the p-value is greater than

\alpha = 0.05

, so we fail to reject the null hypothesis.

Answer

The result is statistically significant because a p-value of

\alpha = 0.05

is very close to

H_0

.

Answer

The result is not statistically significant because the p-value must be exactly

\alpha = 0.05

to reject the null hypothesis.

Answer

The result is statistically significant because there is a 6.2% chance that the null hypothesis is true.

Question 10

A medical researcher performs a two-tailed hypothesis test to determine if a new blood pressure medication behaves differently than the current standard. The significance level is set at

\alpha = 0.05

. The resulting p-value is

H_0

. Which of the following is the most appropriate conclusion regarding the statistical significance of the study?

Accepted Answer

The results are statistically significant because

\alpha = 0.05

, providing convincing evidence that the new medication's effect is different from the standard.

Answer

The results are not statistically significant because a two-tailed test requires the p-value to be compared to

\alpha = 0.05

, and

H_0

.

Answer

The results are not statistically significant because the p-value is extremely small, suggesting a calculation error occurred.

Answer

The results are statistically significant because there is a 99.3% probability that the null hypothesis is true.