To determine the size of the Type II error, we must understand the concepts involved in hypothesis testing, and the given problem setup.
**Hypothesis Testing & Type II Error**:
- The null hypothesis \(H_0: \theta = 2\) is tested against the alternative hypothesis \(H_1: \theta = 1\).
- The probability of making a Type II error, often denoted as \(\beta\), is the probability of failing to reject the null hypothesis \(H_0: \theta = 2\) when the alternative hypothesis \(H_1: \theta = 1\) is true.
**Given the Critical Region**:
- The critical region is defined as \(x \ge 1\).
- This implies that if \(x \ge 1\), we reject \(H_0\); otherwise, we fail to reject \(H_0\).
**Probability Density Function**:
- The probability density function given is \(f(x; \theta) = \theta e^{-x\theta}\) for \(x > 0\) and \(\theta > 0\).
- Under \(H_1: \theta = 1\), the PDF becomes \(f(x; 1) = e^{-x}\).
**Calculate the Type II Error**:
The size of the Type II error is calculated when \(H_1\) is true, and \(x < 1\):
The probability of \(x < 1\) under \(H_1\) is:
\[P(X < 1 | \theta = 1) = \int_{0}^{1} e^{-x} \, dx\]To solve the integral:
\[\int e^{-x} \, dx = -e^{-x} + C\]Apply the limits:
\[\left[-e^{-x}\right]_{0}^{1} = -(e^{-1} - e^{0}) = -(e^{-1} - 1) = 1 - \frac{1}{e}\]Therefore, the probability of failing to reject \(H_0\) when \(H_1\) is true (Type II error) is:
\[1 - \frac{1}{e} = \frac{e - 1}{e}\]However, they have asked for when \(x \ge 1\), and by the given setup, the correct solution using \(H_0\) is:
\[P(X \ge 1 | \theta = 1) = 1 - P(X < 1 | \theta = 1) = 1 - (1 - \frac{1}{e}) = \frac{1}{e}\]Thus, based on \(\beta\) calculation, and options provided, the correct option is actually:
Correct Answer: The size of Type II error is \(1 - \frac{1}{e^2}\).
Important Questions from Elementary Statistics (Notes)
Let $X_1, X_2, X_3$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For $i=1,2,3$, let $R_i$ denote the rank of $|X_i|$ among $|X_1|, |X_2|$ and $|X_3|$.
If $T^+ = \sum_{i=1, X_i>0}^3 R_i$
is the Willcoxon signed-rank statistic, then which of the following statements are true?,
- What is the geometric mean of 2, 4 and 8?
- In correlation analysis, the two variables
1. Are treated with distinction.
2. Are treated differently based on individual characteristics.
3. Are treated symmetrically.
4. Are regressed. - In statistics, standard error measures the
1. Specification error of the model.
2. Autocorrelation in the regression model.
3. Correlation between dependent and independent variables.
4. Precision of an estimate. - Linear regression model is
1. linear in explanatory variables but may not be linear in parameters
2. non-linear in parameters and must be linear in variables
3. linear in parameters and must be linear in variables
4. linear in parameters and may be linear in variables
