सरल रैखिक समाश्रयण (Simple linear regression) में अज्ञात... | Agrigyan

Analysis of properties of normal distribution:

• Correct (Not a characteristic): In a general normal distribution $N(\mu, \sigma^2)$, the mean $\mu$ can be any real number ($-\infty \lt \mu \lt \infty$). It is zero only in the special case of the standard normal distribution $N(0, 1)$.
• Incorrect (Is a characteristic): The total area under the probability density curve equals exactly 1 (total probability).
• Incorrect (Is a characteristic): Because the distribution is bell-shaped and unimodal symmetric, $\text{Mean} = \text{Median} = \text{Mode} = \mu$.
• Incorrect (Is a characteristic): The normal distribution curve is perfectly symmetrical about the vertical ordinate at $x = \mu$.

Analysis of probability models:

• Incorrect: Uniform distribution assigns equal probability across all outcomes without binomial compounding.
• Incorrect: Exponential distribution is a continuous distribution modeling time between Poisson events.
• Incorrect: Poisson distribution models rare events over continuous time/space where $n$ is very large and $p$ very small.
• Correct: The scenario satisfies all criteria of a Binomial experiment $B(n=30, p=0.25)$:
• Fixed number of identical trials ($n = 30$).
• Each question trial has only two mutually exclusive outcomes: success (correct guess) or failure (incorrect guess).
• Constant probability of success on each trial ($p = 1/4 = 0.25$).
• Independent guesses between questions.

Analysis of p-value decision criteria:
The significance level is $\alpha = 5\% = 0.05$.
The universal statistical decision rule states:
Reject $H_0$ if $p\text{-value} \le \alpha$.
Fail to reject $H_0$ if $p\text{-value} \gt \alpha$.

• Incorrect: $0.15 \gt 0.05$, fails to reject $H_0$.
• Incorrect: $0.10 \gt 0.05$, fails to reject $H_0$.
• Incorrect: $0.06 \gt 0.05$, fails to reject $H_0$.
• Correct: $0.046 \lt 0.05$, which falls strictly within the critical rejection region, leading to the rejection of the null hypothesis $H_0$.