Critical Values and Rejection Regions: Videos & Practice Problems

Hypothesis testing involves comparing a test statistic to critical values derived from the alpha level to determine if results fall within the rejection region, leading to rejecting or failing to reject the null hypothesis. For left-tailed, right-tailed, and two-tailed tests, critical values correspond to specific tail probabilities. Calculating the test statistic uses the formula $z=\frac{\chi ̄\; -\mu }{\sigma /\sqrt{n}}$. Both p-value and critical value methods yield consistent conclusions, emphasizing the importance of meeting test criteria like sample size for valid inference.

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 actually true. 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 falls below this critical value, it lies 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\(. The rejection region is the area to the right of this critical value. If the test statistic exceeds \)1.64\(, the null hypothesis is rejected; otherwise, it is not.

For two-tailed tests, the significance level α is split equally between the two tails, so each tail has an area of \)\frac{\alpha}{2}\(. With α = 0.05, each tail has an area of 0.025. The critical values are the z-scores corresponding to these tail areas, approximately \)-1.96\( and \)1.96\(. The rejection regions are the areas beyond these critical values on both tails. If the test statistic lies outside the interval \)[-1.96, 1.96]$, the null hypothesis is rejected; if it lies within, the null hypothesis is not rejected.

This method emphasizes understanding the rejection region, which is the set of values for the test statistic that would lead to rejecting the null hypothesis. By comparing the test statistic to the critical value(s), one can determine whether the observed data is sufficiently unusual under the null hypothesis to warrant rejection.

Overall, using critical values in hypothesis testing reinforces the connection between significance levels, tail probabilities, and decision-making in statistical inference. It provides a clear, visualizable criterion for hypothesis testing that complements the p-value approach, enhancing comprehension of statistical evidence and decision thresholds.

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 $\alpha$ 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): $\pm2.33$, 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 $\alpha$ to find the critical value(s), then determine if the given test statistic is in the rejection region.
$\left(B\right)$
$\alpha =0.10;z=-1.53$
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): $\pm1.64$ Test stat IN rejection region.
• D. Test is RIGHT-tailed, Critical value(s): $1.64$, Test stat IN rejection region.

Answer: (B)

Use $\alpha$ 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): $\pm2.11$, Test stat IN rejection region.

Answer: (D)

Use $\alpha$ 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 (α) of 0.01 and a known population standard deviation (σ) of 5 milliliters.

The first step is to establish the hypotheses. 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, indicating a left-tailed test.

Next, calculate the test statistic using the formula for the z-score in hypothesis testing:

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

where 𝑥̄ is the sample mean (352), μ is the hypothesized population mean (355), σ is the population standard deviation (5), and n is the sample size (50). Plugging in the values:

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

For the p-value method, the test statistic is used to find the probability of observing a value as extreme or more extreme than the test statistic under the null hypothesis. Since this is a left-tailed test, the p-value corresponds to the left tail probability of z = -4.2. Using statistical tables or technology, the p-value is approximately 1.1 × 10⁻⁵, which is much smaller than the significance level α = 0.01.

In the critical value method, instead of calculating the p-value, the critical z-value corresponding to α = 0.01 for a left-tailed test is found. This critical value, denoted as \(z_α\), is approximately -2.33. The rejection region consists of all z-values less than -2.33. Since the calculated test statistic z = -4.2 is less than -2.33, it falls within the rejection region.

Both methods lead to the same conclusion: because the p-value is less than α or the test statistic lies in the rejection region, the null hypothesis is rejected. This provides sufficient evidence to support the claim that the mean volume of soda cans is less than 355 milliliters.

Before finalizing the conclusion, it is essential to verify that the assumptions for the hypothesis test are met. The sample size should be sufficiently large (n > 30) or the population distribution should be approximately normal. In this case, with a sample size of 50, the conditions are satisfied, ensuring the validity of the test results.

A coffee shop owner believes the shop’s average daily sales are \$1,200, with a known population standard deviation of \$50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of \$1,230.
$\left(A\right)$ Use $\alpha =0.10$ & the critical value method to test if the true mean daily sales are higher than \$1200.
$H_0:\mu =$ _____ $H_a:\mu$ ______ $\bar{x}=$ ______ $\sigma =$ ______ $n=$ ______ $z=$
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. Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that μ = 1200.
• B. Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$. There is ENOUGH evidence to conclude that μ > 1200.
• C. Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that μ = 1200.
• D. Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to conclude that μ > 1200.

Answer: (B)

A coffee shop owner believes the shop’s average daily sales are \$1,200, with a known population standard deviation of \$50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of \$1,230.
$\left(B\right)$ The owner’s policy is to reward store managers with a yearly bonus for increased sales. Should the owner give this manager with the bonus?

• A. Yes, the manager was able to increase sales.
• B. No, there was no increase in sales.
• C. More info is required.

Answer: (A)

Performing Hypothesis Test with Critical Values Example 1

When testing whether the true proportion of customers preferring a new product packaging differs from a claimed 60%, the process begins by establishing hypotheses. The null hypothesis (H₀) assumes the population proportion p equals 0.6, reflecting the company's claim. The alternative hypothesis (H_a) posits that p is not equal to 0.6, indicating a two-tailed test since the interest lies in any difference, not specifically higher or lower.

Given a sample of 200 customers with 130 preferring the new packaging, the sample proportion (\(\hat{p}\)) is calculated as:

\[\hat{p} = \frac{x}{n} = \frac{130}{200} = 0.65\]

To evaluate the hypothesis, the test statistic is computed using the z-score formula for proportions:

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

where \(p_0 = 0.6\) is the hypothesized population proportion, and \(n = 200\) is the sample size. Substituting the values:

\[z = \frac{0.65 - 0.6}{\sqrt{\frac{0.6 \times 0.4}{200}}} \approx 1.44\]

With a significance level \(\alpha = 0.05\) for a two-tailed test, the critical z-values correspond to the 2.5% tails on each side of the standard normal distribution. These critical values are approximately:

\[z = \pm 1.96\]

The rejection regions lie beyond these critical values in both tails. Since the calculated z-score of 1.44 falls within the range \(-1.96 < z < 1.96\), it does not enter the rejection region. Therefore, there is insufficient evidence to reject the null hypothesis.

This means the data do not provide strong enough support to conclude that the true proportion of customers preferring the new packaging differs from 60%. Understanding this hypothesis testing framework, including setting up null and alternative hypotheses, calculating sample proportions, computing z-scores, and interpreting critical values, is essential for making informed decisions based on sample data in statistics.

Here’s what students ask on this topic:

What is the role of critical values in hypothesis testing?

Critical values serve as thresholds in hypothesis testing that help determine whether to reject the null hypothesis. They are derived from the significance level, alpha (α), and mark the boundaries of the rejection region. If the test statistic falls beyond the critical value in the rejection region, it indicates that the observed data is unlikely under the null hypothesis, leading to its rejection. For example, in a left-tailed test with α = 0.05, the critical value corresponds to the z-score with 5% of the distribution's area to its left. Understanding critical values is essential because they provide a clear cutoff point for decision-making in hypothesis tests, complementing or serving as an alternative to the p-value method.

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

To find critical values, first identify the type of test and the significance level α. For a left-tailed test, the critical value is the z-score with a left-tail area equal to α. For a right-tailed test, it is the z-score with a right-tail area equal to α. In a two-tailed test, the total α is split equally between both tails, so each tail has an area of α/2. You then find the z-scores corresponding to these tail areas. For example, with α = 0.05, a left-tailed test critical value is approximately $z=-1.64$, a right-tailed test critical value is $z=1.64$, and a two-tailed test has critical values at $z=-1.96$ and $z=1.96$. These values can be found using z-tables or statistical software.

What is the rejection region in hypothesis testing and how is it related to critical values?

The rejection region is the part of the sampling distribution where, if the test statistic falls within it, the null hypothesis is rejected. It is defined by the critical value(s) and corresponds to the extreme tail(s) of the distribution with total area equal to the significance level α. For a left-tailed test, the rejection region is to the left of the critical value; for a right-tailed test, it is to the right; and for a two-tailed test, it includes both tails beyond the positive and negative critical values. If the test statistic lies in this region, it suggests that the observed data is unlikely under the null hypothesis, providing evidence to reject it.

How do the critical value method and p-value method compare in hypothesis testing?

Both the critical value and p-value methods are used to decide whether to reject the null hypothesis, and they lead to the same conclusion. The critical value method involves finding the critical value(s) from the significance level α and comparing the test statistic directly to these values. 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 α. If the p-value is less than α, the null hypothesis is rejected. Both methods require the same initial steps, such as calculating the test statistic and setting hypotheses, differing mainly in the comparison step.

How do you calculate the test statistic for hypothesis testing involving means?

The test statistic for hypothesis testing about a population mean is calculated using the formula: $z=\frac{\chi -\mu }{\sigma ∕\sqrt{n}}$, where $\chi$ is the sample mean, $\mu$ is the population mean under the null hypothesis, $\sigma$ is the population standard deviation, and $n$ is the sample size. This z-score measures how many standard errors the sample mean is from the hypothesized population mean. It is used to determine the position of the sample mean within the sampling distribution, which is essential for deciding whether to reject the null hypothesis.

What criteria must be met to perform a valid hypothesis test using critical values?

To perform a valid hypothesis test using critical values, certain assumptions or criteria must be met. Primarily, the sampling distribution of the test statistic should be approximately normal. This is typically ensured if the population is normally distributed or if the sample size $n$ is sufficiently large (usually $n>\; 30$) due to the Central Limit Theorem. Additionally, the population standard deviation $\sigma$ should be known for z-tests. Meeting these criteria ensures that the critical values derived from the normal distribution are appropriate and that the conclusions drawn from the hypothesis test are reliable.