Let's go through each part step by step as a tutor would explain: --- ### **(a) State the distributions of \( X_i \) and \( Y \) in terms of \( \theta \).** - **\( X_i \)**: Each \( X_i \) is an indicator variable: - \( X_i = 1 \) if the \(i\)th component is defective (probability \( \theta \)) - \( X_i = 0 \) if not defective (probability \( 1 - \theta \)) - So, \( X_i \sim \text{Bernoulli}(\theta) \) - **\( Y = \sum_{i=1}^{50} X_i \)**: Sum of 50 independent Bernoulli(\(\theta\)) random variables. - \( Y \sim \text{Binomial}(n=50, p=\theta) \) --- ### **(b) Show that the proposed test is the uniformly most powerful test (UMP).** - The test is to reject \( H_0: \theta = 0.1 \) if \( Y \geq c \). - By the **Neyman-Pearson Lemma**, for simple hypotheses (\( H_0: \theta = 0.1 \) vs \( H_1: \theta = 0.2 \)), the UMP test uses the likelihood ratio: - The likelihood ratio is monotonic in \( Y \), so the test that rejects for large \( Y \) is UMP. - Therefore, this test is the UMP test for these hypotheses. --- ### **(c) If the critical value of the test is \( c = 9 \), what is the level of significance (\( \alpha \)) of the test?** - **Level of significance (\( \alpha \))**: Probability of rejecting \( H_0 \) when \( H_0 \) is true. - \( \alpha = P(Y \geq 9 \mid \theta = 0.1) \) - \( Y \sim \text{Binomial}(50, 0.1) \) - Use binomial tables, Excel, or calculator: - \( \alpha = P(Y \geq 9) = 1 - P(Y \leq 8) \) - Use `=BINOM.DIST(8, 50, 0.1, TRUE)` in Excel to get \( P(Y \leq 8) \) --- ### **(d) If the critical value is \( c = 9 \), what is the power at \( \theta = 0.2 \)?** - **Power**: Probability of correctly rejecting \( H_0 \) when \( \theta = 0.2 \). - \( \text{Power} = P(Y \geq 9 \mid \theta = 0.2) \) - \( Y \sim \text{Binomial}(50, 0.2) \) - Use `=BINOM.DIST(8, 50, 0.2, TRUE)` in Excel, so power is \( 1 - P(Y \leq 8) \) --- ### **(e) Existence of best test at 0.05 significance** - Consider testing \( H_0: \theta=0.1 \) vs \( H_1: \theta=0.2 \) at \( \alpha = 0.05 \): - Find \( c \) such that \( P(Y \geq c \mid \theta=0.1) \leq 0.05 \). - For binomial, sometimes there is **no integer \( c \)** that gives exactly \( \alpha = 0.05 \) due to discreteness; the closest value may be above or below 0.05. - This means a test with **exact** size 0.05 may not exist. --- ### **(e) [Second part] UMP Test Rejects Based on \( \bar{x} \)** - The UMP test, if it exists, rejects \( H_0 \) if \( \bar{x} \geq c \), where \( c \) is chosen so that: - \( P(\bar{X} \geq c \mid \theta=0.1) = \alpha \) - This is equivalent to the previous test, but phrased in terms of the sample mean. --- ### **(f) Refer to part (c) above:** Given: \[ \frac{Y}{50} \sim N\left(\theta, \frac{\theta(1-\theta)}{50}\right) \] \[ Z = \frac{\sqrt{50}\left(\frac{Y}{50} - \theta\right)}{\sqrt{\theta(1-\theta)}} \sim N(0, 1) \] Testing \( H_0: \theta=0.1 \) vs \( H_1: \theta=0.2 \) at level \( \alpha \), reject if \( \frac{Y}{n} \geq c \) (here n = 50, and \( c = \frac{9}{50} \)). --- #### **(i) What is the level of significance (\( \alpha \)) of the test?** - \( \alpha = P\left(\frac{Y}{50} \geq \frac{9}{50} \mid \theta=0.1\right) \) - This is the same as \( P(Y \geq 9 \mid \theta=0.1) \), **same as in part (c)**. --- #### **(ii) Why is your answer the same or different?** - It is **the same** because testing based on \( Y \) or \( \frac{Y}{n} \) is equivalent; the critical value just gets rescaled. Both are monotonic transformations and lead to the same probability calculation. --- ## **Summary Table** | Part | Key Point | |------|-----------| | (a) | \( X_i \sim \text{Bernoulli}(\theta) \), \( Y \sim \text{Binomial}(50, \theta) \) | | (b) | Neyman-Pearson Lemma: test is UMP | | (c) | \( \alpha = P(Y \geq 9 \mid \theta=0.1) \), use binomial table/Excel | | (d) | Power = \( P(Y \geq 9 \mid \theta=0.2) \), use binomial table/Excel | | (e) | Exact 0.05 test may not exist due to discreteness | | (e) | UMP test: reject if \( \bar{X} \geq c \) | | (f)(i) | \( \alpha = P(Y \geq 9 \mid \theta=0.1) \) | | (f)(ii) | Same as in (c) since tests are equivalent | --- **If you want to see how to use Excel or a calculator to get the exact values for \( \alpha \) or power, let me know!**