Test Statistics & p-Values - College Statistics

1. 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?
   Answer: 0.01 < p-value < 0.02 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.
2. 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 $\alpha=0.01$?
   Answer: $z \approx 2.21$; p-value $\approx 0.027$; Fail to reject $H_0$. 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.
3. A hypothesis test for the fairness of a coin ($H_0: p=0.5$) results in a p-value of 0.86. Which statement provides the best interpretation of this result?
   Answer: The observed data are highly consistent with the null hypothesis, so we fail to reject it. A large p-value means that the observed data are not surprising if the null hypothesis were true. It indicates a lack of evidence against $H_0$. However, failing to find evidence against $H_0$ is not the same as proving $H_0$ is true. The language 'fail to reject' is used to reflect this. Choice C provides the most accurate and nuanced interpretation. Choice A incorrectly states the p-value is the probability $H_0$ is true. Choice B overstates the conclusion. Choice D misinterprets the conditional nature of the p-value.
4. 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?
   Answer: The test statistic and p-value are fixed properties of the sample; changing the decision threshold does not alter the evidence. 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.
5. 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?
   Answer: The p-value is less than 0.01. 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.
6. 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?
   Answer: 0.030 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: $p\text{-value} = 2 \times 0.015 = 0.030$.
7. 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?
   Answer: The t-statistic would be larger, and the p-value would be smaller. 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.
8. 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?
   Answer: 0.05 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.
9. 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?
   Answer: The p-value is less than 0.05, and we reject the null hypothesis. 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.
10. 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?
    Answer: The decision to conduct a one-tailed test was made after observing the data, which invalidates the p-value's standard interpretation. 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.
11. 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: $t_{.025}=2.145$, $t_{.01}=2.624$)
    Answer: The p-value is between 0.01 and 0.025. 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.
12. 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?
    Answer: $t = -2.00$; Reject $H_0$ because the p-value is less than 0.05. 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.
13. 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?
    Answer: 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. 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.
14. 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 $z_B$?
    Answer: $z_B = 2 \times z_A$ 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, $z_B = 2 \times z_A$.
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?
    Answer: An approximately uniform distribution between 0 and 1. When the null hypothesis is true, any p-value is equally likely. The probability of obtaining a p-value less than or equal to $x$ is $x$. 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].
16. 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 $t_{.01}=2.539$).
    Answer: $H_a: \mu > 30$ with $\alpha=0.05$ 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.
17. 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?
    Answer: Study 2 has a smaller t-statistic and a larger p-value. 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.
18. 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 $p_B$?
    Answer: $p_B$ is substantially less than half of $p_A$. 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 $p_A$.
19. 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?
    Answer: The result is statistically significant, but the observed effect size is small and may not be practically meaningful. 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.

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