Name: hypothesis tests using critical values
Uploaded: 2023-04-30T12:24:55.000Z
Duration: 1 min 14 s
Channel: JoVE

K12 Standards

Virginia  Standards of Learning

HIgh School

Physical Science

PS.1

PS.1.a

Decision Making: Traditional Method

The traditional or classical method involves using the critical value to conclude the hypothesis testing.
As a first step, a hypothesis is stated and expressed symbolically as follows.
For the proportion, mean, or standard deviation of a population, the null and alternative hypotheses are expressed as follows.
Further, a critical value is obtained for the chosen parameter in the hypotheses at a specific predetermined significance level &#945;. For proportion, mean, or standard deviation, these critical values at &#945; are the&#160;z,&#160;t,&#160;or chi-square&#160;values, respectively, which are calculated using the&#160;z,&#160;t, or chi-square&#160;distributions.
The critical value is then plotted to demarcate the critical region in the probability distribution.
Further, the test statistic is calculated using the sample data and plotted on the probability distribution curve.
The null hypothesis is rejected when the test statistic value falls within the critical region. However, we fail to reject it when the test statistic falls outside the critical region.

decision making: traditional method

The process of hypothesis testing based on the traditional method includes calculating the critical value, testing the value of the test statistic using the sample data, and interpreting these values.
First, a specific claim about the population parameter is decided based on the research question and is stated in a simple form. Further, an opposing statement to this claim is also stated. These statements can act as null and alternative hypotheses, out of which a null hypothesis would be a neutral statement while the alternative hypothesis can have a direction. The alternative hypothesis can also be the original claim if it involves a specific direction of the parameter.
Once the hypotheses are stated, they are expressed symbolically. As a convention, the null hypothesis would contain the equality symbol, while the alternative hypothesis may contain &#62;, &#60;, or &#8800; symbols.
Before proceeding with hypothesis testing, an appropriate significance level must be decided. There is a general convention of choosing a&#160; 95% (i.e., 0.95) or 99% (i.e., 0.99) level. Here the &#945;&#160;would be 0.05 or 0.01, respectively.
Next, identify an appropriate test statistic. The proportion and mean (when population standard deviation is known) z statistic is preferred. For the mean, when population standard deviation is unknown, it is a t statistic, and for variance (or SD), it is a chi-square statistic.
Then, Calculate the critical value at the given significance level for the test statistic and plot the sampling distribution to observe the critical region. The critical value can be obtained from the z, t, and chi-square tables or electronically using statistical software.
Check if the test statistic falls within the critical region. If it falls within the critical region, reject the null hypothesis.&#160;
The decision about the claim about the property of the population or the general interpretation in this method does not require the P-value.

The process of hypothesis testing based on the traditional method includes calculating the critical value, testing the value of the test statistic using ...

Education

JoVE Core

Statistics

Unit 1

Hypothesis Testing

KEY TERMS AND CONCEPTS

hypothesis tests using critical values

Hypothesis tests using critical values follow a traditional decision process in statistics. Students first state a claim about a population parameter and then write the opposite statement. These become the null hypothesis and the alternative hypothesis. The null hypothesis is neutral, while the alternative hypothesis can show a direction.

After the hypotheses are set, they are written in symbols. By convention, the null hypothesis includes the equality symbol. The alternative hypothesis may use &gt;, &lt;, or ≠, depending on the claim. If the original claim has a specific direction, it can become the alternative hypothesis.

The next step is choosing a significance level. A common choice is 95%, which means α = 0.05, or 99%, which means α = 0.01. The test statistic is chosen after that. For a proportion, or for a mean when the population standard deviation is known, a z statistic is used. For a mean when the population standard deviation is unknown, a t statistic is used. For variance or standard deviation, the test statistic is chi-square.

Then the critical value is calculated for the chosen significance level. This value can be found in z, t, or chi-square tables, or with statistical software. The sampling distribution is plotted to show the critical region. If the test statistic falls in that region, the null hypothesis is rejected. This traditional method makes the decision without using a P-value.

Hypothesis Tests Using Critical Values

what is a hypothesis?

Testing a Hypothesis with Sample Data

A hypothesis is a statement about a population property or a phenomenon seen in an observation or experiment. It can describe something that is expected or predicted for a group. In statistics, it is a claim that can be tested with data. A hypothesis should be general, but not vague. It should not make a claim with a definite number, quantity, or measurement for a population property. A hypothesis is not judged as right or wrong on its own. It must be examined through data collection and an...

null and alternative hypotheses

Comparing Hypotheses in Statistical Tests

Statistical hypothesis testing starts by comparing two opposing claims. These are the null hypothesis and the alternative hypothesis. They give researchers a way to judge sample data before making a decision. The null hypothesis, written as H0, says there is no difference between the variables. It also says the variables are not related. This idea is often treated as the status quo. The alternative hypothesis, written as H1 or Ha, makes a claim that conflicts with H0. It is the statement the...

critical region, critical values and significance level

critical-region-critical-values-and-significance-level

Rejection Regions and Significance in Tests

Rejection regions and significance level help decide whether to reject the null hypothesis in hypothesis testing. A sample statistic is first turned into a test statistic using the z, t, or chi-square distribution. The critical region is the part of the probability curve where the test statistic leads to rejection of the null hypothesis. This region is also called the rejection region or region of rejection. It can appear in the right tail, left tail, or both tails of the distribution. The...

p-value

P-Value in Hypothesis Testing

P-value is a key idea in statistics and hypothesis testing. It stands for probability value. It helps show how likely the observed sample results are if the null hypothesis is true. More exactly, p-value is the probability that another randomly selected sample will produce results as extreme or more extreme than the results from the given sample, assuming the null hypothesis is true. A large p-value suggests that the null hypothesis should not be rejected. However, a larger p-value does not...

types of hypothesis testing

Tail Direction in Hypothesis Tests

Hypothesis tests are classified by the direction of the alternative hypothesis. The null hypothesis is a neutral statement, and the alternative hypothesis is the claim being tested against it. If the null hypothesis says p = 0.5, the alternative can be written as p &gt; 0.5, p 0.5, the critical region lies on the right side of the probability distribution curve. A right-tailed hypothesis test is used in that case. When the claim is p &lt; 0.5, the critical region lies on the left side of the...

decision making: p-value method

Hypothesis Test Results with P-Values

Hypothesis testing with the P-value method uses sample data to judge a claim about a population parameter. The process starts with a clear research question and a simple statement about the population. An opposing statement is also written so the claim can be tested fairly. These statements become the null hypothesis and the alternative hypothesis. The null hypothesis is a neutral statement, while the alternative hypothesis can have a direction. In some cases, the original claim is the...

Hypothesis tests using critical values follow a traditional decision process in statistics. Students first state a claim about a population parameter and then write the opposite statement. These become the null hypothesis and the alternative hypothesis. The null hypothesis is neutral, while the alternative hypothesis can show a direction. After the hypotheses are set, they are written in symbols. By convention, the null hypothesis includes the equality symbol. The alternative hypothesis may...

hypothesis: accept or fail to reject?

Fail to Reject the Null Hypothesis

Hypothesis testing helps scientists decide whether to reject or fail to reject the null hypothesis. The decision comes from the data, the test statistic, the confidence level, the critical values, and the P-value. These parts work together to show whether the evidence is strong enough to support the alternative hypothesis. When the evidence is weak, it is not correct to say that the null hypothesis has been accepted as true. In statistics, that wording can suggest the null hypothesis has been...

errors in hypothesis tests

Type I, Type II, and Test Power

Hypothesis testing can lead to four outcomes based on the true state of the null hypothesis and the choice to reject or not reject it. If the null hypothesis is true and the decision is not to reject it, that is a correct decision. If the null hypothesis is true but it is rejected, that is a Type I error. A Type II error happens when the null hypothesis is false, but it is not rejected. This is also an incorrect decision. The correct decision when the null hypothesis is false is to reject it.

testing a claim about population proportion

testing-a-claim-about-population-proportion

Population Proportion Tests with z and Binomial

Population proportion tests use hypothesis testing to check a claim about a binomial result. The data must come from random and unbiased samples. The variable must also have only two outcomes, such as success or failure, selected or not selected, or true or false. One common method uses the sample proportion and a normal approximation to the binomial distribution. This approach works when the sample is large enough, the proportion is close to 0.5, and np is greater than 5. Here, p is the...

testing a claim about mean: known population sd

testing-a-claim-about-mean-known-population-sd

Z Test for Mean with Known SD

The z test for a population mean uses a known population standard deviation and a normal distribution. It is used when random samples are collected without sampling bias. The sample size should be greater than 30, and the population standard deviation must already be known. In many real situations, the population standard deviation is unknown. In the rare cases when it is known, the claim about the population mean can be tested with the normality assumption and the z distribution. The null...

testing a claim about mean: unknown population sd

testing-a-claim-about-mean-unknown-population-sd

Using the t Test for a Population Mean

Testing a population mean with an unknown standard deviation uses the t test. This method helps estimate a mean when the population standard deviation is not known. It is common in real situations, so the sample standard deviation is used instead. The data should come from randomly selected samples with no sampling bias. The samples should also be approximately normally distributed. There is no specific requirement for sample size, but a different approach is used when the sample size is less...

testing a claim about standard deviation

Chi-Square Test for Standard Deviation Claims

Chi-square hypothesis testing is used to check a claim about population standard deviation or population variance. The data and samples must be random and unbiased. The population distribution must also be normal. The test starts by stating the null and alternative hypotheses clearly and in symbolic form. The claim usually gives a specific value for the population standard deviation or variance. A sample standard deviation or variance is then used to calculate the test statistic. The critical...

key terms and concepts

Learn how the traditional hypothesis testing method uses null and alternative hypotheses, significance levels, test statistics, and critical values to make a decision.

Hypothesis Tests Using Critical Values | JoVE K12

statistics

JoVE Core Statistics introduces the concepts and application of the collection, analysis, presentation, and interpretation of data to students. These animated lessons provide an understanding of statistics beneficial to students from a wide range of disciplines. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide

Physics

PH.1

PH.1.a

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First, a ...

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• independently and collaboratively plan and conduct observational and experimental investigations; identify variables, constants, and controls where appropriate and include the safe use of chemicals and equipment
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• take metric measurements using appropriate tools and technologies
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PS.1.b

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• construct and interpret data tables showing independent and dependent variables, repeated trials, and means
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• apply mathematical concepts and processes to scientific questions
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PS.1.c

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The student will demonstrate an understanding of scientific and engineering practices by constructing and critiquing conclusions and explanations
• construct scientific explanations based on valid and reliable evidence obtained from sources (including the students' own investigations)
• construct arguments supported by empirical evidence and scientific reasoning
• generate and compare multiple solutions to problems based on how well they meet the criteria and constraints
• differentiate between a scientific hypothesis, theory, and law

PS.1.d

The student will demonstrate an understanding of scientific and engineering practices by developing and using models
• construct, develop, and use models and simulations to illustrate and/or explain observable and unobservable phenomena
• evaluate limitations of models

PS.1.e

The Scientific Method

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The student will demonstrate an understanding of scientific and engineering practices by obtaining, evaluating, and communicating information
• read scientific texts, including those adapted for classroom use, to determine the central idea and/or obtain scientific and/or technical information
• gather, read, and synthesize information from multiple appropriate sources and assess the credibility, accuracy, and possible bias of each publication
• construct, use, and/or present an oral and written argument supported by empirical evidence and scientific reasoning

PS.1.f

Hypothesis Tests Using Critical Values

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