Let $X_1, X_2, ..., X_7$ be a random sample drawn from a continuous distribution with unknown unique median $M$. The null hypothesis $H_0: M = 2$ is tested against the alternative $H_1: M \gt 2$ at level of significance 0.05 using the right-tailed test based on the Sign test statistic $K$, which is the number of observations in the sample greater than 2. If the observed sample is $-3, -6, 1, 9, 4, 10, 12$, which of the following statements are true?
The problem involves using the Sign test for a random sample of size n=7 drawn from a continuous distribution to test a hypothesis about the median M.
Sign Test Statistic Calculation
The sample observations are: $-3, -6, 1, 9, 4, 10, 12$.
The null hypothesis is \(H_0: M = 2\) and the alternative is \(H_1: M > 2\).
The test statistic K is defined as the number of observations greater than 2.
- Observations greater than 2 are: $9, 4, 10, 12$.
- The observed value of the test statistic is \(K_{obs} = 4\).
Hypothesis Testing Decision
Under the null hypothesis \(H_0: M = 2\), for a continuous distribution with a unique median, \(P(X > 2) = 0.5\). Thus, the test statistic K follows a Binomial distribution \(B(n=7, p=0.5)\).
The test is a right-tailed test at a significance level \(\alpha = 0.05\).
We need to find the critical value \(k_{crit}\) such that \(P(K \ge k_{crit}) \le 0.05\).
- \(P(K \ge 7) = \binom{7}{7} (0.5)^7 = 1/128 \approx 0.0078\)
- \(P(K \ge 6) = P(K=6) + P(K=7) = \binom{7}{6} (0.5)^7 + \binom{7}{7} (0.5)^7 = (7+1)/128 = 8/128 = 0.0625\)
Since \(P(K \ge 6) > 0.05\) and \(P(K \ge 7) \le 0.05\), the critical value is \(k_{crit} = 7\).
Comparing the observed statistic (\(K_{obs} = 4\)) with the critical value (\(k_{crit} = 7\)): since $4 < 7$, we do not reject \(H_0\). Therefore, statement 1 ('$H_0$ is rejected') is false.
P-value Calculation
The p-value for a right-tailed test is calculated as \(P(K \ge K_{obs})\).
- \(p\text{-value} = P(K \ge 4) = \sum_{k=4}^{7} P(K=k)\)
- \(p\text{-value} = P(K=4) + P(K=5) + P(K=6) + P(K=7)\)
- \(p\text{-value} = \left( \binom{7}{4} + \binom{7}{5} + \binom{7}{6} + \binom{7}{7} \right) (0.5)^7\)
- \(p\text{-value} = (35 + 21 + 7 + 1) / 128 = 64 / 128 = 0.5\)
The p-value is 0.5. Since $0.5 > 0.01$, statement 2 ('The p-value of the test is greater than 0.01') is true.
Expected Value of K
Under the null hypothesis, K follows a Binomial distribution \(B(n=7, p=0.5)\).
The expected value E(K) is calculated using the formula \(E(K) = n \times p\):
$E(K) = 7 \times 0.5 = 3.5$
Therefore, statement 3 ('Under $H_0$, $E(K) = 3.5$') is true.
Confidence Interval Analysis
For a sample size n=7, the interval \([X_{(k)}, X_{(n-k+1)}]\) forms a confidence interval for the median M with a confidence coefficient of \(1 - 2 P(K \le k-1)\), where K ~ B(7, 0.5).
We evaluate the confidence coefficient for the interval \([X_{(2)}, X_{(6)}]\). Here, $k=2$ and $n=7$.
- The interval corresponds to $k=2$, leading to $n-k+1 = 7-2+1 = 6$. The interval is indeed \([X_{(2)}, X_{(6)}]\).
- The confidence coefficient is calculated as \(1 - 2 P(K \le k-1) = 1 - 2 P(K \le 2-1) = 1 - 2 P(K \le 1)\).
- We calculate \(P(K \le 1) = P(K=0) + P(K=1) = \binom{7}{0}(0.5)^7 + \binom{7}{1}(0.5)^7 = (1 + 7)/128 = 8/128 = 1/16 = 0.0625\).
- Confidence Coefficient = \(1 - 2 \times 0.0625 = 1 - 0.125 = 0.875\).
Since the confidence coefficient $0.875$ is less than $0.95$, statement 4 ('$[X_{(2)}, X_{(6)}]$ is a confidence interval for $M$ with confidence coefficient at least 0.95') is false.
Conclusion
Based on the analysis of each statement:
- Statement 1 ('$H_0$ is rejected') is false.
- Statement 2 ('The p-value of the test is greater than 0.01') is true.
- Statement 3 ('Under $H_0$, $E(K) = 3.5$') is true.
- Statement 4 ('$[X_{(2)}, X_{(6)}]$ is a confidence interval for $M$ with confidence coefficient at least 0.95') is false.
Therefore, the statements regarding the p-value being greater than 0.01 and the expected value being 3.5 are true.
Important Questions from Random Variables
Let $X_1$ and $X_2$ be a random sample from Uniform$[0, \theta]$ distribution, where $\theta > 0$. For testing the hypothesis
$H_0: \theta = 1$ against $H_1: \theta = 2$,
consider a test which rejects $H_0$ if $X_1 + X_2 > \frac{4}{5}$. Then, the probability of type-I error is- Let $\{Y_n: n \ge 1\}$ be a sequence of independent and identically distributed random variables, where $Y_1 \sim \text{Bernoulli}(\frac{1}{2})$. Define $Z = \sum_{n=1}^\infty \frac{4Y_n}{5^n}$. Then, which of the following statements is true?
Let $X$ be a single sample from an absolutely continuous distribution with probability density function
$f(x|\theta) = \begin{cases} \frac{2}{\theta^2}(\theta - x), & \text{if } 0 < x < \theta \\ 0, & \text{otherwise,} \end{cases}$
where $\theta > 0$ is unknown. Which of the following intervals is a $95\%$ confidence interval for $\theta$?- Let $X$ and $Y$ be two independent random variables such that the moment generating func-tions of $X$ and $Y$ are
$M_X(t) = e^{3(e^t-1)}, t \in R$,
and
$M_Y(t) = (\frac{1}{2}e^{-3t} + \frac{1}{2}e^{3t})^2, t \in R$,
respectively. Then which of the following statements are true? - $X_1, X_2, \cdots$ are independent identically distributed random variables having common density $f$. Assume $f(x) = f(-x)$ for all $x \in \mathbb{R}$. Which of the following statements is correct?
