College Statistics Quiz: Test Statistics And P Values

20 questions · exam conditions

0:00

Test Statistics And P ValuesQuestion 1 of 20

A study on commute times results in a test statistic of $t = 2.50$ from a sample of size $n=20$ . The null hypothesis is $H_0: \mu = 30$ minutes. For which of the following alternative hypotheses and significance levels would the null hypothesis be rejected? (For df=19, $t_{.025}=2.093$ and $t_{.01}=2.539$ ).

H_a: \mu \neq 30

with

\alpha=0.02

H_a: \mu < 30

with

\alpha=0.05

H_a: \mu > 30

with

\alpha=0.05

H_a: \mu > 30

with

\alpha=0.01

← Back to quizzes

College Statistics Quiz

College Statistics Quiz: Test Statistics And P Values

Practice Test Statistics And P Values in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Test Statistics And P Values, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$H_a: \mu \neq 30$ with $\alpha=0.02$
$H_a: \mu < 30$ with $\alpha=0.05$
$H_a: \mu > 30$ with $\alpha=0.05$ (correct answer)
$H_a: \mu > 30$ with $\alpha=0.01$

Explanation: With

t=2.50

and

df=19

: For

H_a: \mu > 30

(right-tailed), the p-value is

P(t_{19} > 2.50)

. Since

2.093 < 2.50 < 2.539

, the p-value is between 0.01 and 0.025. This p-value is less than 0.05, so we would reject

H_0

for choice C. For choice D, we would fail to reject because the p-value is > 0.01. For

H_a: \mu < 30

(left-tailed), the p-value is very large (

P(t_{19} < 2.50) > 0.975

), so we would not reject. For

H_a: \mu \neq 30

(two-tailed), the p-value is

2 \times P(t_{19} > 2.50)

, so it is between 0.02 and 0.05. This p-value is not less than

\alpha=0.02

(choice A), so we would fail to reject.

Question 2

In a test comparing the means of two independent groups, with $H_a: \mu_1 \neq \mu_2$ , a researcher calculates a test statistic of $t = -2.70$ with 20 degrees of freedom. Given the t-distribution critical values $t_{.01, df=20}=2.528$ and $t_{.005, df=20}=2.845$ , which of the following correctly identifies the range for the p-value?

0.01 < p-value < 0.02 (correct answer)
0.005 < p-value < 0.01
0.02 < p-value < 0.05
The p-value is less than 0.005.

Explanation: When you encounter a two-tailed t-test with given critical values, you need to determine where your test statistic falls relative to these benchmarks to find the p-value range. Since this is a two-tailed test (

H_a: \mu_1 \neq \mu_2

), you're looking at both tails of the distribution. Your test statistic is

t = -2.70

, but since we're dealing with a two-tailed test, what matters is the absolute value:

|t| = 2.70

. Now compare this to the given critical values:

t_{.01, df=20} = 2.528

and

t_{.005, df=20} = 2.845

. Since

2.528 < 2.70 < 2.845

, your test statistic falls between these two critical values. The critical value

t_{.01}

corresponds to an alpha level of 0.01 for a two-tailed test, meaning each tail contains 0.005. Similarly,

t_{.005}

corresponds to an alpha of 0.005 for a two-tailed test. Since your test statistic exceeds

t_{.01}

but doesn't reach

t_{.005}

, the p-value must be between 0.01 and 0.02. Choice A is correct: 0.01 < p-value < 0.02. Choice B (0.005 < p-value < 0.01) places the p-value too low—your test statistic isn't extreme enough. Choice C (0.02 < p-value < 0.05) suggests the result is less significant than it actually is. Choice D claims the p-value is less than 0.005, but your test statistic doesn't exceed the

t_{.005}

threshold. Study tip: Always remember that in two-tailed tests, the critical values given correspond to the total alpha level, so compare your test statistic's absolute value to these benchmarks to bracket the p-value.

Question 3

In a two-tailed hypothesis test for a population proportion, a data analyst calculates the test statistic as $z = 2.17$ . Which of the following is the closest p-value for this test?

0.015
0.030 (correct answer)
0.970
0.985

Explanation: The p-value for a two-tailed test is the probability of observing a test statistic as extreme or more extreme in either direction. This is calculated as

2 \times P(Z > |z_{obs}|)

. For

z = 2.17

, the area in the upper tail is

P(Z > 2.17) \approx 0.015

. Since the test is two-tailed, this area must be doubled:

p\text{-value} = 2 \times 0.015 = 0.030

Question 4

A coffee shop manager claims that the average customer wait time is less than 3 minutes. A random sample of 36 customers has a mean wait time of 2.8 minutes with a sample standard deviation of 0.6 minutes. A hypothesis test is conducted to evaluate the manager's claim. What is the test statistic, and what is the resulting conclusion at the $\alpha = 0.05$ significance level?

$t = -2.00$ ; Fail to reject $H_0$ because the p-value is greater than 0.05.
$t = 2.00$ ; Reject $H_0$ because the p-value is less than 0.05.
$t = -0.33$ ; Fail to reject $H_0$ because the test statistic is small.
$t = -2.00$ ; Reject $H_0$ because the p-value is less than 0.05. (correct answer)

Explanation: When you encounter a hypothesis test about a population mean with unknown population standard deviation, you're dealing with a one-sample t-test. The manager's claim that average wait time is "less than 3 minutes" signals a left-tailed test. First, set up your hypotheses:

H_0: \mu = 3

and

H_a: \mu < 3

. The test statistic formula is

t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}

. Substituting the values:

t = \frac{2.8 - 3}{0.6/\sqrt{36}} = \frac{-0.2}{0.1} = -2.00

. With 35 degrees of freedom (n-1 = 35) and a left-tailed test at

\alpha = 0.05

, the critical value is approximately -1.69. Since -2.00 < -1.69, we reject

H_0

. The p-value for t = -2.00 is approximately 0.027, which is less than 0.05. Choice A calculates the correct test statistic but reaches the wrong conclusion—it fails to recognize that a p-value of 0.027 is indeed less than 0.05. Choice B has the correct magnitude but wrong sign for the test statistic; since the sample mean (2.8) is less than the hypothesized mean (3), the test statistic must be negative. Choice C makes a calculation error, possibly confusing the numerator and denominator or using wrong values entirely. Choice D correctly identifies both the test statistic (-2.00) and the proper conclusion (reject

H_0

). Remember: the sign of your test statistic should align with your sample data relative to the null hypothesis, and always compare your p-value to the significance level carefully.

Question 5

A hypothesis test results in a p-value of 0.06. The pre-specified significance level was $\alpha=0.05$ , so the researcher initially fails to reject $H_0$ . A colleague suggests that if the significance level were changed to $\alpha=0.10$ , the result would be significant. Which statement is the most statistically sound response to this suggestion?

Changing the significance level is acceptable, so the conclusion should be changed to rejecting $H_0$ .
The test statistic and p-value depend on the alpha level, so they must be recalculated before concluding.
The test statistic and p-value are fixed properties of the sample; changing the decision threshold does not alter the evidence. (correct answer)
The original test was flawed since the p-value was so close to the alpha level; the study should be discarded.

Explanation: The p-value is calculated from the sample data and is independent of the significance level,

\alpha

. It represents the strength of the evidence against the null hypothesis. The alpha level is a pre-determined threshold for making a decision. While changing the threshold would change the conclusion, it does not change the evidence (the p-value) itself. Changing

\alpha

after seeing the p-value is poor scientific practice.

Question 6

A pilot study with $n=50$ participants is conducted to test $H_0: \mu = 100$ . The study finds a sample mean of 105 and a sample standard deviation of 25, yielding a test statistic of $t=1.41$ . A follow-up study is then conducted which finds the exact same sample mean (105) and sample standard deviation (25), but with a much larger sample size of $n=200$ . How would the t-statistic and p-value of the follow-up study most likely compare to the pilot study?

The t-statistic would be smaller, and the p-value would be larger.
The t-statistic would be larger, and the p-value would be smaller. (correct answer)
The t-statistic and p-value would remain the same because the sample mean and standard deviation are the same.
The t-statistic would be larger, and the p-value would also be larger.

Explanation: The formula for the t-statistic is

t = (\bar{x} - \mu_0) / (s/\sqrt{n})

. Since

\bar{x}

\mu_0

, and

s

are constant, the t-statistic is proportional to

\sqrt{n}

. The sample size quadrupled from 50 to 200, so the new t-statistic will be

\sqrt{4}=2

times the original t-statistic (

1.41 \times 2 = 2.82

). A larger (more extreme) t-statistic corresponds to a smaller p-value, indicating stronger evidence against the null hypothesis.

Question 7

After collecting data, an analyst performs a two-tailed test for a population mean and finds a p-value of 0.09. They notice that the sample mean is in the direction they originally suspected. They then perform a right-tailed test with the same data and find a p-value of 0.045. They conclude the result is significant at $\alpha=0.05$ . Why is this conclusion flawed?

The p-value for a one-tailed test should be one-fourth that of a two-tailed test, indicating a calculation error.
The analyst should have used the original two-tailed p-value of 0.09, which is less than $\alpha=0.10$ and therefore significant.
A p-value of 0.045 is not considered statistically significant at the conventional $\alpha=0.05$ level.
The decision to conduct a one-tailed test was made after observing the data, which invalidates the p-value's standard interpretation. (correct answer)

Explanation: When you encounter hypothesis testing questions involving changes in test direction, the key issue is whether the decision was predetermined or data-driven. This question tests your understanding of proper hypothesis testing protocol and the dangers of "p-hacking." The analyst's conclusion is fundamentally flawed because they chose to switch from a two-tailed to a one-tailed test after seeing that the sample mean supported their original suspicion. This post-hoc decision invalidates the statistical interpretation. Proper hypothesis testing requires that you specify your test type (one-tailed or two-tailed) before collecting and analyzing data. When you change your analysis strategy based on what the data shows, you're essentially "shopping" for a significant result, which inflates the probability of Type I error beyond your stated

\alpha

level. Looking at the wrong answers: Choice A incorrectly states the mathematical relationship between one-tailed and two-tailed p-values. The correct relationship is that a one-tailed p-value should be half (not one-fourth) of a two-tailed p-value when testing in the correct direction. Choice B misses the point entirely by focusing on a different significance level rather than addressing the procedural flaw. Choice C is factually wrong—a p-value of 0.045 is indeed less than

\alpha = 0.05

and would typically be considered significant. Remember this key principle: your hypothesis and test type must be determined before data analysis begins. Any post-hoc changes to your statistical approach based on preliminary results compromise the validity of your conclusions and constitute poor statistical practice.

Question 8

Two independent studies test the hypothesis $H_0: \mu=100$ . Both studies use a sample size of $n=49$ and find a sample mean of $\bar{x}=104$ . Study 1 reports a sample standard deviation of $s_1=14$ , while Study 2 reports a sample standard deviation of $s_2=21$ . How do the test statistic ( $t$ ) and p-value from Study 2 compare to those from Study 1?

Study 2 has a larger t-statistic and a smaller p-value.
Study 2 has a smaller t-statistic and a larger p-value. (correct answer)
Study 2 has a smaller t-statistic and a smaller p-value.
The t-statistics and p-values are identical because the means and sample sizes are the same.

Explanation: The t-statistic is calculated as

t = (\bar{x} - \mu_0) / (s/\sqrt{n})

. The numerator

\bar{x} - \mu_0 = 4

and denominator term

\sqrt{n} = 7

are the same for both studies. Thus, the t-statistic is inversely proportional to the sample standard deviation

s

. Since Study 2 has a larger standard deviation (

s_2 > s_1

), it will have a smaller t-statistic. A smaller t-statistic is less extreme, corresponding to a larger p-value.

Question 9

For a right-tailed z-test of a population mean, the critical value for a significance level of $\alpha=0.05$ is $z_{crit} = 1.645$ . If a researcher calculates a test statistic from their sample data and finds it to be exactly $z_{obs} = 1.645$ , what is the p-value of the test?

0.05 (correct answer)
0.025
0.10
The p-value cannot be determined without the sample size.

Explanation: When you encounter hypothesis testing problems involving critical values and test statistics, you're dealing with the fundamental relationship between these values and p-values. The key insight is understanding what happens when your test statistic exactly equals the critical value. In a right-tailed z-test, the critical value

z_{crit} = 1.645

represents the z-score that cuts off the upper 5% of the standard normal distribution (since

\alpha = 0.05

). The p-value represents the probability of observing a test statistic as extreme or more extreme than what you actually observed, assuming the null hypothesis is true. Since your observed test statistic

z_{obs} = 1.645

exactly equals the critical value, the p-value is the area to the right of 1.645 under the standard normal curve. This area is precisely 0.05, making answer A correct. Answer B (0.025) would be incorrect because this represents half of the significance level, which might arise from confusion with two-tailed tests where

\alpha

is split between both tails. Answer C (0.10) is twice the correct value and doesn't correspond to any meaningful relationship here. Answer D is wrong because p-values in z-tests depend only on the test statistic and the direction of the test, not the sample size directly. Remember this key relationship: when your test statistic exactly equals the critical value, the p-value equals the significance level (

\alpha

). This makes intuitive sense because you're right at the boundary of the rejection region.

Question 10

A chi-square goodness-of-fit test is performed on a set of categorical data with 5 categories, resulting in a test statistic of $\chi^2 = 10.5$ . For a significance level of $\alpha = 0.05$ , the critical value is 9.49. Which of the following is a correct conclusion?

The p-value is less than 0.05, and we fail to reject the null hypothesis.
The p-value is greater than 0.05, and we fail to reject the null hypothesis.
The p-value is less than 0.05, and we reject the null hypothesis. (correct answer)
The degrees of freedom should be 5, not 4, so no conclusion can be drawn.

Explanation: The degrees of freedom for a goodness-of-fit test are

k-1

, where

k

is the number of categories. Here,

df = 5-1=4

. The decision rule using the critical value method is to reject

H_0

if the test statistic is greater than the critical value. Since

\chi^2_{obs} = 10.5 > 9.49

, we reject

H_0

. A test statistic that exceeds the

\alpha=0.05

critical value must have a p-value less than 0.05. Therefore, both parts of statement C are correct.

Question 11

A medical researcher conducts a randomized controlled trial to test a new drug. They test the hypothesis $H_0: \mu_{new} = \mu_{placebo}$ against $H_a: \mu_{new} > \mu_{placebo}$ and obtain a p-value of 0.022. What is the correct interpretation of this p-value?

The probability that the new drug is ineffective (i.e., the null hypothesis is true) is exactly 0.022.
If the new drug were truly ineffective, the probability of observing a sample difference as large or larger than what was found is 0.022. (correct answer)
The probability that a randomly selected patient will experience the observed treatment effect is 0.022.
There is a 97.8% probability that the alternative hypothesis is true and the drug is effective.

Explanation: The p-value is the probability of obtaining a test result at least as extreme as the one observed, under the assumption that the null hypothesis is true. Choice B is the correct definition applied to this context. Choice A is a common misinterpretation; the p-value is not the probability of the null hypothesis being true. Choice D is the '1 - p-value' fallacy. Choice C confuses the p-value with an individual outcome probability.

Question 12

A researcher testing $H_0: \mu=50$ against $H_a: \mu < 50$ collects a sample of $n=15$ items and calculates a test statistic of $t=-2.50$ . Based on the t-distribution with 14 degrees of freedom, which of the following is the most accurate statement about the p-value? (Relevant critical values for df=14: $t_{.025}=2.145$ , $t_{.01}=2.624$ )

The p-value is less than 0.01.
The p-value is between 0.01 and 0.025. (correct answer)
The p-value is between 0.025 and 0.05.
The p-value is greater than 0.05.

Explanation: The test is a one-sided, left-tailed test. The p-value is the area to the left of

t=-2.50

on a t-distribution with

df=15-1=14

. Due to symmetry, this is equal to the area to the right of

t=2.50

. The given critical values show that

t=2.50

is between

t_{.025}=2.145

and

t_{.01}=2.624

. Therefore, the area in the tail is between 0.01 and 0.025.

Question 13

Researcher A conducts a study and obtains a test statistic of $z_A = 2$ . Researcher B conducts a similar study and obtains a test statistic of $z_B = 4$ . Let $p_A$ and $p_B$ be the corresponding two-tailed p-values. Which statement accurately describes the relationship between $p_A$ and $p_B$ ?

$p_B$ is approximately half of $p_A$ .
$p_B$ is greater than $p_A$ .
$p_B$ is substantially less than half of $p_A$ . (correct answer)
$p_B$ is only slightly smaller than $p_A$ .

Explanation: The p-value is determined by the area in the tails of the standard normal distribution. For

z_A=2

, the two-tailed p-value is

2 \times P(Z>2) \approx 0.0455

. For

z_B=4

, the two-tailed p-value is

2 \times P(Z>4) \approx 0.000063

. The p-value decreases much more rapidly than the z-statistic increases because the normal distribution's tails are very thin. Therefore,

p_B

is much smaller than half of

p_A

Question 14

A polling agency wants to test if a candidate's approval rating is different from 50%. In a random sample of 1,000 voters, 535 express approval. They test the hypotheses $H_0: p = 0.5$ vs. $H_a: p \neq 0.5$ . What are the test statistic and p-value, and what is the conclusion at $\alpha=0.01$ ?

$z \approx 2.21$ ; p-value $\approx 0.027$ ; Fail to reject $H_0$ . (correct answer)
$z \approx 2.21$ ; p-value $\approx 0.027$ ; Reject $H_0$ .
$z \approx 1.11$ ; p-value $\approx 0.267$ ; Fail to reject $H_0$ .
$z \approx 2.21$ ; p-value $\approx 0.014$ ; Reject $H_0$ .

Explanation: When you encounter hypothesis testing problems about proportions, you need to calculate the test statistic, find the p-value, and compare it to your significance level to make a conclusion. For this one-sample z-test for proportions, the test statistic is

z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}

. Here,

\hat{p} = \frac{535}{1000} = 0.535

p_0 = 0.5

, and

n = 1000

. This gives us

z = \frac{0.535 - 0.5}{\sqrt{\frac{0.5(0.5)}{1000}}} = \frac{0.035}{\sqrt{0.00025}} = \frac{0.035}{0.0158} \approx 2.21

. Since this is a two-tailed test (

H_a: p \neq 0.5

), the p-value is

2 \times P(Z > 2.21) \approx 2 \times 0.0136 \approx 0.027

. With

\alpha = 0.01

, since

0.027 > 0.01

, we fail to reject

H_0

. Choice A correctly identifies all these values and the proper conclusion. Choice B makes the same calculations but incorrectly rejects

H_0

– this represents the common error of rejecting when p-value > α. Choice C has an incorrect test statistic of 1.11, possibly from a calculation error in the standard error. Choice D has the right test statistic but wrong p-value (0.014 instead of 0.027), likely from using a one-tailed test or rounding errors. Study tip: Always double-check whether your test is one-tailed or two-tailed, as this directly affects your p-value calculation. For two-tailed tests, remember to multiply by 2, and always compare your p-value to α carefully before making conclusions.

Question 15

An experiment is designed to test a null hypothesis $H_0$ that is, in reality, true. If this experiment were repeated thousands of times with new random samples each time, what would be the approximate distribution of the resulting p-values?

A normal distribution centered at 0.50.
A distribution skewed to the right, with most p-values close to 0.
A distribution skewed to the left, with most p-values close to 1.
An approximately uniform distribution between 0 and 1. (correct answer)

Explanation: When the null hypothesis is true, any p-value is equally likely. The probability of obtaining a p-value less than or equal to

x

x

. For example, there's a 5% chance of getting a p-value of 0.05 or less, a 10% chance of getting a p-value of 0.10 or less, and so on. This property defines a uniform distribution on the interval [0, 1].

Question 16

A statistician calculates a 99% confidence interval for a population mean $\mu$ and finds it to be [120, 150]. The team now wishes to perform a two-tailed hypothesis test of $H_0: \mu = 115$ against $H_a: \mu \neq 115$ . What can be concluded about the p-value for this test?

The p-value is less than 0.01. (correct answer)
The p-value is exactly 0.01.
The p-value is between 0.01 and 0.05.
The p-value is greater than 0.05.

Explanation: There is a direct correspondence between a two-sided hypothesis test and a confidence interval. A

C

% confidence interval contains all the values of

\mu_0

for which we would not reject the null hypothesis

H_0: \mu = \mu_0

at a significance level of

\alpha = 1 - C/100

. Here, a 99% confidence interval corresponds to an alpha level of

\alpha = 0.01

. Since the null value

\mu_0 = 115

is outside the interval [120, 150], the null hypothesis would be rejected at

\alpha = 0.01

. This implies that the p-value for the test must be less than 0.01.

Question 17

With a very large sample of 100,000 individuals, a study finds that a new diet plan results in an average weight loss of 0.5 pounds more than an old plan over one year. The difference is statistically significant with a p-value of < 0.0001. What is the most appropriate conclusion?

The new diet plan is proven to be significantly better and should be widely recommended.
The p-value is extremely small, indicating a very large and important treatment effect.
The result is statistically significant, but the observed effect size is small and may not be practically meaningful. (correct answer)
The study must be flawed because a small effect like 0.5 pounds cannot have such a small p-value.

Explanation: This question highlights the difference between statistical significance and practical significance. A very large sample size can make even a tiny, unimportant effect statistically significant (i.e., produce a very small p-value). While the data provide strong evidence that a difference exists, the magnitude of that difference (0.5 pounds over a year) is very small and may have no practical importance. A small p-value does not imply a large effect size.

Question 18

Two separate political polls test the hypothesis $H_0: p=0.6$ . Poll A uses a sample of $n_A = 400$ people. Poll B uses a sample of $n_B = 1600$ people. Both polls happen to find the exact same sample proportion, $\hat{p}=0.65$ . Let $z_A$ and $z_B$ be the test statistics for Poll A and Poll B, respectively. What is the relationship between $z_A$ and $z_B$ ?

$z_B = 0.25 \times z_A$
$z_B = 0.5 \times z_A$
$z_B = 2 \times z_A$ (correct answer)
$z_B = 4 \times z_A$

Explanation: The formula for the one-proportion z-test statistic is

z = (\hat{p} - p_0) / \sqrt{p_0(1-p_0)/n}

. The numerator,

\hat{p} - p_0

, is the same for both polls. The denominator contains

\sqrt{n}

. Thus, the z-statistic is proportional to

\sqrt{n}

. Since the sample size for Poll B is four times the sample size for Poll A (

n_B = 4n_A

), its test statistic will be

\sqrt{4}=2

times larger. Therefore,

z_B = 2 \times z_A

Question 19

In a two-tailed hypothesis test for a population proportion, a data analyst calculates the test statistic as $z = 2.17$ . Which of the following is the closest p-value for this test?

0.015
0.030 (correct answer)
0.970
0.985

Explanation: The p-value for a two-tailed test is the probability of observing a test statistic as extreme or more extreme in either direction. This is calculated as

2 \times P(Z > |z_{obs}|)

. For

z = 2.17

, the area in the upper tail is

P(Z > 2.17) \approx 0.015

. Since the test is two-tailed, this area must be doubled:

p\text{-value} = 2 \times 0.015 = 0.030

Question 20

The t-statistic would be smaller, and the p-value would be larger.
The t-statistic would be larger, and the p-value would be smaller. (correct answer)
The t-statistic and p-value would remain the same because the sample mean and standard deviation are the same.
The t-statistic would be larger, and the p-value would also be larger.

Explanation: The formula for the t-statistic is

t = (\bar{x} - \mu_0) / (s/\sqrt{n})

. Since

\bar{x}

\mu_0

, and

s

are constant, the t-statistic is proportional to

\sqrt{n}

. The sample size quadrupled from 50 to 200, so the new t-statistic will be

\sqrt{4}=2

times the original t-statistic (

1.41 \times 2 = 2.82

). A larger (more extreme) t-statistic corresponds to a smaller p-value, indicating stronger evidence against the null hypothesis.