Name: population proportion tests with z and binomial
Uploaded: 2023-04-30T12:25:10.000Z
Duration: 1 min 24 s
Channel: JoVE

In the natural populations of Trinidadian guppies, females select the males with orange coloration for mating.
To determine if guppy populations in an aquarium also show the same behavior, an experiment is conducted where 12 females are individually introduced to three orange males and three blue males simultaneously.
It is originally claimed that the females choose the orange males.
So, the null hypothesis would state that an equal number of females would show a preference for orange and blue males. The alternative hypothesis is that a higher number of females would prefer the orange males.
The experiment shows that ten out of twelve females preferred the orange males.
This ratio provides the sample proportion&#8212; 0.83&#8212;which is used to get the test statistic as follows.
It is observed that this test statistic falls within the critical region at a significance level of 0.05.
Also, the P-value from this z statistic is 0.011.
So, we may conclude that the aquarium population of guppies shows the same mating preference as observed in the natural population.

testing a claim about population proportion

Testing a Claim about Population Proportion

A complete procedure for testing a claim about a population proportion is provided here.
There are two methods of testing a claim about a population proportion: (1) Using the sample proportion from the data where a binomial distribution is approximated to the normal distribution and (2) Using the binomial probabilities calculated from the data.
The first method uses normal distribution as an approximation to the binomial distribution. The requirements are as follows: sample size is large enough, the probability of proportion p is close to 0.5, the np (product of sample size and proportion) is greater than 5, and the critical values can be calculated using the z distribution. It also requires the samples to be random and unbiased, and the nature of the data to be&#160; binomial, i.e., there are only two possible outcomes (e.g., success or failure; selected or not selected, true or false, etc.). A proportion is binomial in nature. So, this method is well suitable for testing a claim using hypothesis testing for population proportion.
As a first step, the hypotheses (null and alternative hypotheses) are stated clearly and expressed symbolically. The proportion p used in hypotheses statements is the assumed proportion value, often 0.5. The proportion obtained from the data is the sample proportion. Both these values are crucial in calculating the z statistic.
The critical value can then be obtained from the z distribution utilizing&#160; the normal approximation of the binomial distribution. The critical value can be positive or negative based on the hypothesis direction; accordingly, the hypothesis test is right-tailed, left-tailed, or two-tailed. The critical value is calculated at any desired confidence level, most commonly 95% or 99%.
The P-value is then directly calculated using the z statistic and the critical z value, and the hypothesis test is concluded. The z statistic can also be directly compared with the critical value to conclude the hypothesis test.
The second method of testing the claim about proportion does not require np &#62; 5 as it uses the exact binomial distribution without normal approximation. This method does not calculate the critical value. Instead, it uses the probabilities of obtaining&#160; x (the value of successes out of total trials, e.g., 60 successes out of 110 trials) in the n trials. It calculates the probabilities of x or fewer and x or greater and then leads to the&#160; P-values. This second method of testing a claim about proportion is tedious to do manually and requires statistical software. Nonetheless, the inferences determined in both ways are equally accurate.

A complete procedure for testing a claim about a population proportion is provided here.
There are two methods of testing a claim about a population ...

testing-a-claim-about-population-proportion

Education

JoVE Core

Statistics

Unit 1

Hypothesis Testing

KEY TERMS AND CONCEPTS

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 assumed proportion used in the hypotheses, often 0.5, while the sample proportion comes from the data.

The first step is to write the null and alternative hypotheses in symbolic form. Then the z statistic is calculated from the sample proportion and the claimed proportion. A critical value is found from the z distribution at a chosen confidence level, often 95% or 99%.

The direction of the claim determines whether the test is right-tailed, left-tailed, or two-tailed. The critical value may be positive or negative. The decision can be made by comparing the z statistic with the critical value or by using the P-value from the z test.

A second method uses the exact binomial distribution instead of a normal approximation. It does not require np &gt; 5. This method works with the probability of getting x successes in n trials, such as 60 successes out of 110 trials.

For the exact method, the probabilities of x or fewer and x or greater are calculated and used to find the P-value. It does not use a critical value, and it is tedious to do by hand, so statistical software is usually needed. Even so, both methods give equally accurate conclusions.

Population Proportion Tests with z and Binomial

population-proportion-tests-with-z-and-binomial

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.

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 to test a claim about a population proportion using the z test or exact binomial probabilities, with clear steps for hypothesis testing and P-values.

Population Proportion Tests with z and Binomial | 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

Population Proportion Tests with z and Binomial

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