Name: chi-square test for standard deviation claims
Uploaded: 2023-04-30T12:25:26.000Z
Duration: 1 min 19 s
Channel: JoVE

Correct pricing of gold requires an accurate scale, and its accuracy is achieved by reducing the standard deviation of the mean weight.
Consider an example where a company claims that they have significantly reduced the standard deviation of their scales from 0.005 g to 0.003 g tested over 30 individual units.
To test this claim, a hypothesis test is conducted where the null hypothesis states that the old and improved models have an equal standard deviation. The alternative hypothesis states that the improved model is more accurate and has a significantly smaller standard deviation than the old model.
Testing the hypothesis requires the sample statistic to be converted to the &#935;2 statistic as follows.
Here, the critical region at a 0.05 significance level falls at the left tail of the curve.
Observe that the &#935;2 value calculated from the sample falls within it.
Also, the P-value obtained using the left-tailed test is less than 0.05.
So, the improved model proves significantly more accurate than the old model based on the test result.

testing a claim about standard deviation

Testing a Claim about Standard Deviation

A complete procedure to test a claim about population standard deviation or population variance is explained here.
The hypothesis testing for the claim of population standard deviation (or variance) requires the data and samples to be random and unbiased. The population distribution also must be normal. There is no specific requirement on the sample size as the estimation is based on the chi-square distribution.
As a first step, the hypothesis (null and alternative) concerning the claim about population SD (or variance) should be stated clearly and expressed symbolically. The hypothesis generally claim a certain value of SD or variance to be tested. Samples provide sample SD or variance. Using both these values, the test statistic is calculated.
The critical value here depends on the sample size (or the degrees of freedom) calculated from the chi-square distribution. Based on the directionality in the alternative hypothesis, the test can be left-tailed, right-tailed, or two-tailed. The sample test statistic is compared with the critical chi-square value generally calculated at a 95% or 99% confidence level. Otherwise, P-value is obtained and compared with the significance level of 0.05 or 0.01 to conclude the hypothesis test.

A complete procedure to test a claim about population standard deviation or population variance is explained here.
The hypothesis testing for the claim of ...

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JoVE Core

Statistics

Unit 1

Hypothesis Testing

KEY TERMS AND CONCEPTS

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 value comes from the chi-square distribution and depends on the sample size, or more exactly, the degrees of freedom. The test can be left-tailed, right-tailed, or two-tailed, depending on the direction of the alternative hypothesis. The sample test statistic is then compared with the critical chi-square value, usually at a 95% or 99% confidence level.

A P-value can also be used instead of a critical value approach. In that case, the P-value is compared with the significance level, such as 0.05 or 0.01. The result is used to conclude the hypothesis test for the claim about standard deviation or variance.

Chi-Square Test for Standard Deviation Claims

chi-square-test-for-standard-deviation-claims

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...

decision making: traditional method

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...

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

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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...

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 chi-square hypothesis testing checks a claim about population standard deviation or variance using sample data, critical values, and P-values.

Chi-Square Test for Standard Deviation Claims | JoVE K12

statistics

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Chi-Square Test for Standard Deviation Claims

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