Name: using the t test for a population mean
Uploaded: 2023-04-30T12:49:25.000Z
Duration: 1 min 21 s
Channel: JoVE

Males of two species of Fan-throated lizards flap their dewlaps differently: the pale-colored species flaps its dewlap at a faster rate than the tricolored species.
Field observations of 32 individuals show that the pale-colored species flaps the dewlap on an average of 10 times per unit time while the tricolored species flaps it on an average of 6 times per unit time.
To test the claim, we begin with the null hypothesis that there is no difference between the dewlap flapping rate in both species. In contrast, the alternative hypothesis states that the pale-colored species flaps the dewlap at a faster rate, which is the original claim.
As the population standard deviation of the dewlap flapping rate is unknown, this hypothesis test is conducted using the t distribution for which the sample standard deviation is utilized to calculate the test statistic.
Here, the test statistic can be seen within the critical region at the right tail.
Additionally, the P-value from the calculated t statistic is less than 0.05 supporting the original claim.

testing a claim about mean: unknown population sd

Testing a Claim about Mean: Unknown Population SD

A complete procedure of testing a hypothesis about a population mean when the population standard deviation is unknown is explained here.
Estimating a population mean requires the samples to be approximately normally distributed. The data should be collected from the randomly selected samples having no sampling bias. There is no specific requirement for sample size. But if the sample size is less than 30, and we don&#39;t know the population standard deviation, a different approach is used; instead of the z distribution, the t distribution is used for calculating the test statistic and critical value.
As in most realistic situations, the population standard deviation is often unknown; testing the claim about the population mean would utilize the sample standard deviation. The critical value is calculated using the t distribution (at specific degrees of freedom calculated from sample size) instead of the z distribution.
The hypothesis (null and alternative) should be stated clearly and then expressed symbolically. The null hypothesis is a neutral statement stating population mean is equal to some definite value. The alternative hypothesis can be based on the mean claimed in the hypothesis with an inequality sign. The right-tailed, left-tailed, or two-tailed hypothesis test can be decided based on the sign used in the alternative hypothesis.
As the method does not require normal distribution, the critical value is calculated using the t distribution (t table). It is generally calculated at 95% or 99% of the desired confidence level. As per the traditional method, the sample t statistic calculated from the sample data is compared with the t score (t critical value) obtained from the t table. The P-value is calculated based on the data as per the P-value method. Both these methods help conclude the hypothesis test.

A complete procedure of testing a hypothesis about a population mean when the population standard deviation is unknown is explained here.
Estimating a ...

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

Education

JoVE Core

Statistics

Unit 1

Hypothesis Testing

KEY TERMS AND CONCEPTS

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 than 30 and the population standard deviation is unknown.

In that case, the t distribution replaces the z distribution. The t distribution is used to find both the test statistic and the critical value. The critical value depends on the degrees of freedom, which are calculated from the sample size.

The hypothesis test starts with clear null and alternative hypotheses. The null hypothesis is a neutral statement that the population mean equals a specific value. The alternative hypothesis uses an inequality sign and may lead to a right-tailed, left-tailed, or two-tailed test.

The critical value is found with a t table, usually at the 95% or 99% confidence level. The sample t statistic is then compared with the t critical value in the traditional method. The P-value method can also be used. Both methods support the final conclusion about the claim.

Using the t Test for a Population Mean

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

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 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 to test a claim about a population mean when the standard deviation is unknown using the t distribution, t table, and P-value method.

Using the t Test for a Population Mean | 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

Using the t Test for a Population Mean

Description

Transcript

Tags