4.15 Central limit theorem Quiz

1. $X\sim N(50,\ 16)$. A random sample of $4$ is taken. What is the distribution of $\bar{X}$?

• A. $N(50,\ 4)$
• B. $N(50,\ 16)$
• C. $N(200,\ 64)$
• D. $N(50,\ 2)$

2. $X$ and $Y$ are independent, $\mathrm{Var}(X)=9$ and $\mathrm{Var}(Y)=4$. Find $\mathrm{Var}(X-Y)$.

• A. $5$
• B. $13$
• C. $25$
• D. $1$

3. A skewed population has mean $20$ and standard deviation $6$. For a random sample of $36$, what is $\mathrm{P}(\bar{X}>21)$ (to 3 s.f.)?

• A. $0.309$
• B. $0.0668$
• C. $0.159$
• D. $0.0227$

4. Which statement about the central limit theorem is correct?

• A. It makes any single observation normal.
• B. It needs the population to be normal.
• C. It applies to sample means from large samples.
• D. It applies only when $n<30$.

5. A random sample of $n=100$ is taken from a population with standard deviation $15$. What is the standard deviation of $\bar{X}$?

• A. $150$
• B. $1.5$
• C. $15$
• D. $0.15$