Critical Values and Rejection Regions: Videos & Practice Problems

Hypothesis testing involves comparing a test statistic, such as a z-score, to critical values derived from the significance level $\alpha$. For left-tail, right-tail, or two-tail tests, critical values define rejection regions where the null hypothesis is rejected if the test statistic falls within these regions. The critical value method parallels the p-value approach, both assessing evidence against the null hypothesis. Calculating the test statistic uses sample data and population parameters, ensuring criteria like sample size are met for valid inferential statistics conclusions.

Critical Values and Rejection Regions

In hypothesis testing, critical values serve as thresholds that distinguish between expected test statistics and unusual ones, providing an alternative to the p-value method. Both approaches involve comparing a test statistic to a benchmark, but while the p-value method compares the p-value to the significance level (α), the critical value method compares the test statistic directly to critical values derived from α.

To find critical values, start with the significance level α, which represents the probability of rejecting the null hypothesis when it is true (Type I error). For a left-tailed test, the critical value corresponds to the z-score where the left tail area equals α. For example, with α = 0.05, the critical value is approximately \(-1.64\). If the test statistic is less than this critical value, it falls in the rejection region, leading to rejection of the null hypothesis.

In a right-tailed test, the critical value is the z-score where the right tail area equals α. Using the same α = 0.05, the critical value is about \$1.64\(. The rejection region lies to the right of this critical value, so if the test statistic exceeds \)1.64\(, the null hypothesis is rejected.

Two-tailed tests split the significance level between both tails, so each tail has an area of \)\frac{\alpha}{2}\(. For α = 0.05, each tail has an area of 0.025. The critical values are the z-scores at these tail probabilities, approximately \)-1.96\( and \)1.96\(. The rejection regions are both tails beyond these critical values. If the test statistic lies outside the interval \)[-1.96, 1.96]$, the null hypothesis is rejected.

This method emphasizes understanding the relationship between the significance level, critical values, and rejection regions. The critical value approach provides a clear visual and numerical boundary for decision-making in hypothesis testing, reinforcing the concept that rejecting or failing to reject the null hypothesis depends on whether the test statistic falls within or beyond these critical thresholds.

Overall, mastering the use of critical values enhances comprehension of hypothesis testing by linking the significance level directly to the test statistic's position, offering a complementary perspective to the p-value method.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$\left(A\right)$ The test statistic & the critical value are the same thing.

• A. TRUE
• B. FALSE
• C. Cannot be determined.

Answer: (B)

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$\left(B\right)$ The critical value is the boundary of the rejection region.

• A. TRUE
• B. FALSE
• C. Cannot be determined

Answer: (A)

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$\left(C\right)$ You should always reject $H_0$ if the test statistic is greater than the critical value.

• A. TRUE
• B. FALSE
• C. Cannot be determined

Answer: (B)

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.
$\left(A\right)$ $H_0:\mu =7.5$; $H_a:\mu >7.5$
$\alpha =0.01;z=2.17$

Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

• A. Test is TWO-tailed, Critical value(s): $\pm 2.33$, Test stat IN rejection region.
• B. Test is LEFT-tailed, Critical value(s): $-2.33$, Test stat NOT IN rejection region.
• C. Test is RIGHT-tailed, Critical value(s): $2.33$, Test stat NOT IN rejection region.
• D. Test is RIGHT-tailed, Critical value(s): $2.17$, Test stat IN rejection region.

Answer: (C)

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.
$\left(B\right)$ $H_0:p=0.64$;
$\alpha =0.10;z=-1.53$

Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

• A. Test is LEFT-tailed, Critical value(s): $-1.28$, Test stat NOT IN rejection region.
• B. Test is LEFT-tailed, Critical value(s): $-1.28$, Test stat IN rejection region.
  
• C. Test is TWO-tailed, Critical value(s): $\pm 1.64$ Test stat IN rejection region.
• D. Test is RIGHT-tailed, Critical value(s): $1.64$, Test stat IN rejection region.

Answer: (B)

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.
$\left(C\right)$ $H_0:\mu =98.6$; $H_a:\mu \ne 98.6$
$\alpha =0.05$
$t=2.41;df=17$

Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

• A. Test is RIGHT-tailed, Critical value(s): $2.11$, Test stat IN rejection region.
• B. Test is RIGHT-tailed, Critical value(s): $2.11$, Test stat NOT IN rejection region.
• C. Test is RIGHT-tailed, Critical value(s): $0.025$, Test stat IN rejection region.
• D. Test is TWO-tailed, Critical value(s): $\pm 2.11$, Test stat IN rejection region.

Answer: (D)

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.
$\left(D\right)$ $H_0:{\sigma }^2=0.3;H_a:{\sigma }^2>0.3$
$\alpha =0.05$
${\chi }^2=61.7;df=51$

Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

• A. Test is RIGHT-tailed, Critical value(s): $68.7$, Test stat IN rejection region.
• B. Test is RIGHT-tailed, Critical value(s): $68.7$, Test stat NOT IN rejection region.
• C. Test is RIGHT-tailed, Critical value(s): $61.7$, Test stat NOT IN rejection region.
• D. Test is LEFT-tailed, Critical value(s): $-61.7$, Test stat NOT IN rejection region.

Answer: (B)

Performing Hypothesis Test with Critical Values

When conducting a hypothesis test, both the critical value method and the p-value method follow similar steps to determine whether to reject or fail to reject the null hypothesis. Consider a scenario where a soda company claims its cans contain an average of 355 milliliters, but a sample of 50 cans shows a mean volume of 352 milliliters. The goal is to test if the mean volume is less than the claimed amount, using a significance level of α = 0.01 and a known population standard deviation σ = 5.

First, the hypotheses are established: the null hypothesis (H₀) states that the population mean μ equals 355 milliliters, while the alternative hypothesis (H₁) posits that μ is less than 355 milliliters. This sets up a left-tailed test because the alternative hypothesis uses a less-than sign.

Next, the test statistic is calculated using the z-score formula for a population mean when the population standard deviation is known:

\[z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]

Here, the sample mean \(\bar{x}\) is 352, the population mean μ is 355, σ is 5, and the sample size n is 50. Plugging in these values yields:

\[z = \frac{352 - 355}{5 / \sqrt{50}} \approx -4.2\]

At this stage, the two methods diverge. Using the p-value method, the z-score is used to find the probability of observing a value as extreme or more extreme than the test statistic under the null hypothesis. For a left-tailed test, this is the cumulative probability to the left of z = -4.2, which is approximately \$1.1 \times 10^{-5}\(. Since this p-value is much smaller than the significance level α = 0.01, it provides strong evidence against the null hypothesis.

Alternatively, the critical value method involves finding the z critical value corresponding to the significance level α for a left-tailed test. This critical value, denoted as \)z_{\alpha}\(, satisfies:

\[P(Z < z_{\alpha}) = \alpha\]

For α = 0.01, the critical value is approximately \)z_{\alpha} = -2.33$. The rejection region consists of all z-scores less than -2.33. Since the calculated test statistic z = -4.2 is less than -2.33, it falls within the rejection region, leading to rejection of the null hypothesis.

Both methods ultimately lead to the same conclusion: there is sufficient evidence to reject the null hypothesis and support the claim that the mean volume is less than 355 milliliters. It is important to verify that the assumptions for the hypothesis test are met, such as the sample size being sufficiently large (n > 30) or the population distribution being normal. In this case, with n = 50, the sample size criterion is satisfied, ensuring the validity of the test results.

Understanding how to apply both the critical value and p-value methods enhances the ability to interpret hypothesis tests effectively, reinforcing the connection between test statistics, significance levels, and decision-making in inferential statistics.

A soup company claims that the average sodium content of their most popular soup is 500 mg per can. A nutritionist collects a sample of 36 cans with mean sodium content 507 mg. Assume a known pop. standard deviation of 15 mg & test the nutritionist’s suspicion that the mean sodium content is more than 500 mg using the critical value method with $\alpha =0.01$

$H_0:\mu =$ ____ $H_a:\mu$ ______
$\bar{x}=$ ______ $\sigma =$ ______ $n=$ ______
$z=$
Critical Value(s):
Because test stat. $\left(z\right)$ is [ INSIDE | OUTSIDE ] rejection region, we [ REJECT | FAIL TO REJECT ] $H_0$. There is [ ENOUGH | NOT ENOUGH ] evidence to conclude that…

• A. Critical Value(s): $2.8$ Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that $\mu>500$
• B. Critical Value(s): $2.8$ Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that $\mu>500$
• C. Critical Value(s): $2.33$ Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that $\mu >500$
• D. Critical Value(s): $2.33$ Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that $\mu>500$

Answer: (C)

Performing Hypothesis Test with Critical Values Example 1

When testing a population proportion, such as the claim that 40% of households have backyard gardens, it is essential to establish clear hypotheses. The null hypothesis (H₀) assumes the population proportion p equals 0.4, reflecting the city's claim. The alternative hypothesis (H_a) tests whether the true proportion differs from 0.4, expressed as p ≠ 0.4, indicating a two-tailed test.

To evaluate this, a sample of 150 households is surveyed, with 72 reporting gardens. The sample proportion, denoted as \(\hat{p}\), is calculated by dividing the number of successes by the sample size:

\[\hat{p} = \frac{x}{n} = \frac{72}{150} = 0.48\]

The test statistic used is a z-score, which measures how many standard deviations the sample proportion is from the hypothesized population proportion. The formula for the z-score in a proportion test is:

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

Substituting the values:

\[z = \frac{0.48 - 0.4}{\sqrt{\frac{0.4 \times 0.6}{150}}} = 2\]

Since the alternative hypothesis is two-sided, the significance level \(\alpha = 0.1\) is split between the two tails of the normal distribution, allocating 0.05 to each tail. The critical z-values corresponding to these tail areas are approximately ±1.64. These critical values define the rejection regions: any test statistic less than -1.64 or greater than 1.64 leads to rejecting the null hypothesis.

Here, the calculated z-score of 2 exceeds the positive critical value of 1.64, placing it in the rejection region. This result provides sufficient evidence to reject the null hypothesis and conclude that the true proportion of households with backyard gardens is significantly different from 40% at the 10% significance level.

Understanding this process highlights the importance of hypothesis formulation, calculation of sample statistics, and interpretation of test statistics within the context of significance levels and critical values. This approach is fundamental in inferential statistics for making data-driven decisions about population parameters.

Here’s what students ask on this topic:

What is the role of critical values in hypothesis testing?

Critical values in hypothesis testing serve as thresholds that separate the range of expected test statistics from unusual ones. They are derived from the significance level, denoted as $\alpha$, and define the rejection regions. If the calculated test statistic falls within these rejection regions, we reject the null hypothesis. For example, in a left-tail test, the critical value corresponds to the z-score where the left tail area equals $\alpha$. This method provides an alternative to the p-value approach by directly comparing the test statistic to the critical value to decide whether to reject or fail to reject the null hypothesis.

How do you find critical values for left-tail, right-tail, and two-tail hypothesis tests?

To find critical values, first identify the type of test: left-tail, right-tail, or two-tail. For a left-tail test, the critical value is the z-score with a left tail area equal to $\alpha$. For a right-tail test, it is the z-score with a right tail area equal to $\alpha$. In a two-tail test, the total area in both tails equals $\alpha$, so each tail has an area of $\alpha /2$. The critical values are the z-scores corresponding to these tail areas, typically found using z-tables or technology. For example, with $\alpha =\; 0.05$, the two-tail critical values are approximately $-1.96$ and $1.96$.

What is the rejection region in hypothesis testing and how is it determined?

The rejection region is the part of the distribution where, if the test statistic falls within it, the null hypothesis is rejected. It is determined by the critical values, which are based on the significance level $\alpha$. For a left-tail test, the rejection region is the area to the left of the critical value. For a right-tail test, it is the area to the right of the critical value. In a two-tail test, the rejection region includes both tails beyond the critical values. The size of the rejection region corresponds to $\alpha$ or $\alpha /2$ in each tail for two-tailed tests. If the test statistic lies in this region, it indicates that the observed data is unlikely under the null hypothesis, leading to its rejection.

How does the critical value method compare to the p-value method in hypothesis testing?

Both the critical value and p-value methods are used to decide whether to reject the null hypothesis, but they approach the decision differently. The critical value method involves finding a threshold value from the significance level $\alpha$ and comparing the test statistic directly to this critical value. If the test statistic falls in the rejection region beyond the critical value, the null hypothesis is rejected. The p-value method calculates the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true, and compares this p-value to $\alpha$. If the p-value is less than $\alpha$, the null hypothesis is rejected. Both methods yield the same conclusion when applied correctly.

What are the criteria for performing a valid hypothesis test using critical values?

To perform a valid hypothesis test using critical values, certain criteria must be met. The sample data should be approximately normally distributed, or the sample size $n$ should be sufficiently large (usually $n>\; 30$) to invoke the Central Limit Theorem. Additionally, the population standard deviation $\sigma$ should be known for z-tests. Meeting these conditions ensures that the test statistic follows the assumed distribution, allowing accurate determination of critical values and valid conclusions about the null hypothesis.

How do you interpret the test statistic in relation to the critical value in hypothesis testing?

The test statistic is compared to the critical value(s) to determine whether it falls within the rejection region. For a left-tail test, if the test statistic is less than the critical value, it lies in the rejection region, and the null hypothesis is rejected. For a right-tail test, if the test statistic is greater than the critical value, it is in the rejection region, leading to rejection of the null hypothesis. In a two-tail test, if the test statistic is less than the lower critical value or greater than the upper critical value, it falls in the rejection region. If the test statistic does not fall in these regions, we fail to reject the null hypothesis, indicating insufficient evidence against it.