Question
Type II error occurs when :
The correct answer is
H₀ (Null hypothesis) is false and is accepted
Type II Error Definition in Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data. It involves evaluating two competing statements about a population parameter:
- The Null Hypothesis (denoted as $H_0$): This is usually a statement of 'no effect' or 'no difference'. It represents the status quo or a baseline assumption.
- The Alternative Hypothesis (denoted as $H_a$ or $H_1$): This is the statement that contradicts the null hypothesis. It represents what we are trying to find evidence for (e.g., an effect, a difference).
Understanding Errors in Hypothesis Testing
When conducting a hypothesis test, there are two possible errors we can make:
- Type I Error (False Positive): This occurs when we reject the null hypothesis ($H_0$) even though it is actually true. The probability of making a Type I error is denoted by alpha ($\alpha$), often called the significance level.
- Type II Error (False Negative): This occurs when we fail to reject the null hypothesis ($H_0$) when it is actually false. This means we miss detecting an effect or difference that truly exists. The probability of making a Type II error is denoted by beta ($\beta$).
Analyzing the Options
Let's look at the outcomes described in the options:
- Option 1: $H_0$ is true and is accepted. This is a correct decision (True Negative).
- Option 2: $H_0$ is true and is rejected. This is a Type I error (False Positive).
- Option 3: $H_0$ is false and is accepted. This is a Type II error (False Negative).
- Option 4: $H_a$ is false and is rejected. If $H_a$ is false, it means $H_0$ is true. Rejecting $H_0$ when it's true is a Type I error. This option is phrased confusingly but equates to rejecting a true $H_0$.
Summary of Hypothesis Testing Outcomes
The possible outcomes of a hypothesis test can be summarized in the following table:
| Decision Based on Sample Data | True State of Null Hypothesis ($H_0$) | |
|---|---|---|
| $H_0$ True | $H_0$ False | |
| Accept $H_0$ | Correct Decision (True Negative) | Type II Error (False Negative) ($\beta$) |
| Reject $H_0$ | Type I Error (False Positive) ($\alpha$) |
Correct Decision (True Positive) |
Conclusion
Based on the standard definitions and the summary table, a Type II error occurs specifically when the null hypothesis ($H_0$) is actually false, but the test results lead us to accept or fail to reject it.
Therefore, the correct description is: $H_0$ (Null hypothesis) is false and is accepted.
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Important Questions from Hypothesis testing - Teaching
- Which of the following is the condition where χ2\chi^2χ2 (chi-square) should not be applied ?
- The null hypothesis that all slope coefficients are simultaneously equal to zero is tested in logit model by:
- The null hypothesis in nonparametric test often _______.
1. Includes specification of a population's parameters
2. Is used to evaluate some general population aspect
3. Is very similar to that used in regression analysis
4. Simultaneously tests more than two population parameters - The _______ test determines whether there is a significant difference between the observed and hypothesized distribution for a sample.
1. Independence
2. Coefficient of determination
3. Correlation analysis
4. Goodness-of-fit - Match the LIST-I with LIST-II
LIST-I LIST-II A. One-Tailed Test I. Null hypothesis is rejected if the sample value is significantly higher or lower than the hypothesized value of the population parameter B. Paired difference Test II. A hypothesis test of the difference between the sample means of two independent samples C. Two-Tailed Test III. A sample value significantly above the hypothesized population value will lead to rejection of the null hypothesis D. Upper-Tailed Test IV. Concerned only with whether the observed value deviates from the hypothesized value in one direction
Choose the correct answer from the options given below:
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