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In statistics, the F-test formula is used in a hypothesis to check if the variances of two different sets of values are equal or not. In other words, the F test is a test statistic with and-distribution under the null hypothesis.
- It is used to compare data sets based on a given or available data collection.
- The data in an F test follows an f distribution.
- This test divides two variances and compares them using the f statistic.
- The formula was coined by George W Snedecor in honour of Sir Ronald A Fisher.
- F-test formula is denoted as- F Value = variances of set 1/ variances of set 2.
σ21 / σ22→ σ2 = ∑ (x−¯x)2 / n-1.
- Here, σ2 is variance, x is the value given in a set of data, ¯x is the mean of the data, n is the total number of values.
- The concept is used in the stock market to determine whether the returns on stocks are equal across two or more portfolios.
Key Terms: F-test Formula, Hypothesis, F-distribution, Variance, Null Hypothesis, Data Sets, F-Test, Statistics, Mean, Population, Standard Deviation, F-Distribution
F-Test Formula
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The f-Test formula is a form of statistical test that is used to compare the variance of two given samples. It is also used to check the ratio of variances between multiple samples.
- F-Test Formula is applied under the null hypothesis.
- The formula uses two degrees of freedom.
- It is used to determine the rejection value using a critical zone.
- To conduct this test, the population should be following f-distribution, and samples must be independent.
- It can either be a one-tailed or two-tailed function.
- The general formula for F-test is given as:
F Value = variances of set 1/ variances of set 2
F = σ21 / σ22
σ2 = ∑ (x−¯x)2 / n-1.
- Where, σ2 is variance, x is the value given in a set of data, ¯x is the mean of the data, n is the total number of values.
The F-Test formula for different set of hypothesis are as follows:
Left Tailed Test
The conditions for left tailed test are as follows:
- Null Hypothesis of test: H0 : σ21= σ22
- Alternate Hypothesis of test: H1 : σ21< σ22
- In case the value of f statistic < f critical value then reject the null hypothesis
Right Tailed Test
The conditions for right tailed test are as follows:
- Null Hypothesis of test: H0 : σ21= σ22
- Alternate Hypothesis of test: H1 : σ21> σ22
- In case the value of f test statistic > f test critical value then reject the null hypothesis
Two Tailed Test
The conditions for two tailed test are as follows:
- Null Hypothesis of test: H0 : σ21= σ22
- Alternate Hypothesis of test: H1 : σ21≠ σ22
- In case the value of f test statistic > f test critical value then reject the null hypothesis
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Steps to calculate F-test formula
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F-test is a test in statistics, that calculate F distribution and compare data sets based on available data. To calculate the F-test formula some steps are to be followed:
- State the null hypothesis with the alternative hypothesis
- Calculate the ‘F’ value with the use of standard formula
- Find the value of F-statistics.
- The ratio of variances of the group of means to the mean of the within-group variances.
- Now, either accept or reject the Null Hypothesis
- The formula for F-distribution is given as:
F Value = variances of set 1/ variances of set 2
F Value = σ21 / σ22
F Test Formula
F-test Formula Equation to Compare Two Variances
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F-Test in statistics compares two variances by dividing them. Since the variances are positive, the result is also a positive number. When the variances are equal, then the ratio of variances is also equal.
- It helps compare the variances of two different sets of values.
- Most importantly, calculate the mean of the two given observations first.
- It is assumed in the process that variance is equal to one.
- Thus proving that variance is equal in the null hypothesis.
Example of F-test Formula Equation to Compare Two VariancesExample- Taking two sets of variances, (variance of 20 )and set 2 = 20, then the ratio is 20/20 =1
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F- Test Statistic Formula Assumptions
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There are various assumptions with the F-test. Some of the important f-test statistic formula assumptions are as follows:
- In the f-test formula, it is easier to conduct the right-tailed test.
- The larger variance should be in the numerator.
- In a two-tailed test, the alpha is divided by two before deriving the right value.
- It is assumed that variance should be squared by standard deviations.
- Use the larger critical value if the degree of freedom is not listed in the F-table.
- It will reduce the possibility of I-Type errors.
Things to Remember
- The F-test formula in statistics is used to compare two variances by dividing them.
- When the variances are equal, then the ratio of variances is also equal.
- The null hypothesis is rejected when the f-value is less than one.
- The critical value is not equal to zero.
- In case the critical value is equal to zero, then the mean of every sample is exactly the same.
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Sample Questions
Ques: What is an F-test formula? (2 marks)
Ans: F test is a test statistic with an F-distribution under the null hypothesis. It compares data sets based on a given or available data collection. Since the variances are positive, the result is also a positive number. F Value = σ21 / σ22
When the variances are equal, then the ratio of variances is also equal.
Ques: How is the standard formula for F-test denoted? (2 marks)
Ans: F-test formula is denoted as-
F Value = variances of set 1/ variances of set 2 → σ21 / σ22
→ σ2 = ∑ (x−¯x)2 / n-1
Here, σ2 is variance, x is the value given in a set of data, ¯x is the mean of the data, n is the total number of values
Ques: What is the right-tailed and two-tailed test? (4 marks)
Ans: The right-tailed is used when the hypothesis statement is greater than the symbol.
Right tailed test- Null hypothesis: H: σ2 = σ
Alternative hypothesis: H: σ22
Result- if, f test statistics > f test critical value
Then reject the null hypothesis
The two-tailed is used when a sample is greater or smaller than a certain range of values.
Two-tailed test- Null hypothesis: H: σ2 = σ22
Alternative hypotheses: H: σ2 ≠ σ22
Result- if , f test statistics > f test critical value
Then reject the null hypothesis
Ques: What are the assumptions of the F-test? (3 marks)
Ans: There are various assumptions with the F-test. The population must be normally distributed. The assumptions are –
- The right-tailed test is easier to calculate.
- The larger variance should be in the numerator.
- In a two-tailed test, the alpha is divided by 2 before deriving the right value.
- The variance should be squared of standard deviations.
- Use the larger critical value, if the degree of freedom is not listed in F-table. It will reduce the possibility of I-Type errors.
Ques: What are the characteristics of F- distribution? (3 marks)
Ans: Some key characteristics of F-distribution are as follows-
- The curve is not symmetrical, skewed to the right
- Each data has a different curve.
- With the increased degree of freedom in the numerator and denominator, the curve is normal
Ques: Mention the steps to calculate the F-test formula? (3 marks)
Ans: F-test is a test in statistics, that calculates F distribution and compares data sets based on available data. To calculate the F-test formula some steps are to be followed-
- State the null hypothesis with the alternative hypothesis
- Calculate the ‘F’ value with the use of standard formula
- Find the value of F-statistics. The ratio of variances of the group of means to the mean of the within-group variances.
Ques: How will you show the F-test formula equation to compare two variances? (3 marks)
Ans: F-test in statistics compare the two variances by dividing them. Since the variances are positive, the result is also a positive number. F Value = σ21 / σ22 When the variances are equal, then the ratio of variances is also equal. Taking two sets of variances,(variance of 20 )and set 2 = 20, then the ratio is 20/20 =1. So, we can see the population is equal. The population variance is always equal when we do F-test. We can say that the variances are equal to 1 with your null hypothesis.
Ques: Ice-cream delivery times of two cities are given below
City 1: Number of delivery times observed = 18, Variance = 38
City 2: Number of delivery times observed = 15, Variance = 83
Check if the delivery times of city 1 is less than city 2 at a 0.05 alpha level? (3 marks)
Ans: This is the case of a left-tailed F test. Thus, the alpha level is 1 - 0.05 = 0.95
- H0: s21=s22
- H1: s21<s22
- Since 38 < 83 thus, city 1 will be sample 1 and city 2 is sample 2.
- n1= 18, n2= 15
- df1= 18 - 1 = 17
- df2 = 15 - 1 = 14
- s21 = 38, s22 = 83
- F = s21/s22 = 38 / 83
- F = 0.4578
- It is given thar F table for 0.95 alpha level is not available, the critical value is determined as follows:
- F(0.95, 17, 14) = 1 / 1.93 = 0.5181
- As 0.4578 < 0.5181 so null hypothesis can be rejected
Ques: What are the conditions for left nailed test? (2 marks)
Ans: The conditions for left nailed test are as follows:
- Null Hypothesis of test: H0 : σ21= σ22
- Alternate Hypothesis of test: H1 : σ21< σ22
- In case the value of f statistic < f critical value then reject the null hypothesis
Ques: What is the difference between f-test and t-test? (3 marks)
Ans: The difference between f-test and t-test are as follows:
| F-Test | T-Test |
|---|---|
| F-test is conducted to check the whether variance of two population are equal. | T-test is conducted when the value of standard deviation and sample size is unknown. |
| The method is used for variance. | The method is used for means. |
| Its formula is σ21/σ22 | Its formula is t = ¯¯¯x−μs/√n |
Ques: Conduct a F-test for the following sets. (3 marks)
a) Set 1 with variance equal to 108 and sample size equal to 31.
b) Set 2 with variance equal to 65 and sample size equal to 21?
Ans: The process is as follows:
- Step 1: The hypothesis statements are as follows:
H-0: No difference in value of variances
H-a: Difference in value of variances
- Step 2: Determine the value of F critical. In this case, the highest variance value is taken as the numerator and the lowest variance value in the denominator.
F-value = σ21/σ22
F-value =108/65
F-value = 1.66
- Step 3: Now calculate degrees of freedom.
The degree of freedom for set 1 is 31 -1 = 30.
The degree of freedom for set 2 is 21 - 1 = 20.
- Step 4: Standard alpha level value is 0.05. During the test, it is reduced to half the initial value, and hence it becomes 0.025.
- Step 5: Use the F table, the critical value is determined with alpha at 0.025. The critical value for (30, 20) at alpha equal to 0.025 is 2.287.
- Step 6: It is now the time for comparing the calculated value with the standard value in the table. It is clear from the values that 1.66 < 2.287. Hence, the null hypothesis is rejected.
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