Central Limit Theorem and Confidence Intervals

Given i.i.d. random variables Y1,Y2,…,Yn with mean μ and variance σ^2, Which of the following statements is true according to the Central Limit Theorem?

• ∑ Yi is exactly normally distributed for any sample size n.

• As n increases, the sum ∑ Yi approaches a normal distribution with mean nμ and variance nσ^2.

• The sum ∑ Yi is uniformly distributed for large n.

• ∑ Yi will be normally distributed only if the original random variables Yi are normally distributed.

Answer explanation

According to CLT, as n increases, the sum ∑ Yi approaches a normal distribution with mean nμ and variance nσ^2.

Which of the following best describes the distribution of the sample mean x̄ when the sample size n is large?

• x̄ ∼ N(μ, σ^2)

• x̄ ∼ N(μ, σ^2 / n)

• x̄ ∼ U (μ, σ^2)

• x̄ ∼ N (nμ, nσ^2)

If x1,x2, ⋯,xn​ are normally distributed with mean μ and variance σ^2, what is the distribution of the sample mean x̄?

• x̄ is exactly normally distributed for any n.

• x̄ is approximately normally distributed only when n is large.

• Uniform distribution with mean μ and variance σ^2/n
• Exponential distribution with mean μ and variance σ^2/n

What does a 95% confidence interval for a population parameter θ represent?

• The interval will contain 95% of all sample values.

• 95% of the population data falls within the interval.

• 95% of the time, the interval contains the true value of θ.

• Range of values within which we are 99% confident the true population parameter lies.

If you increase the confidence level from 95% to 99%, what happens to the width of the confidence interval?

• The width increases.

• The width decreases.

• Remain the same
• The width depends on the sample size only.

When constructing a confidence interval for the mean of a normally distributed population with unknown variance, which distribution is used?

• Standard normal distribution

• binomial distribution
• Student's t-distribution

• exponential distribution

Suppose you calculate a 95% confidence interval for the difference between two independent sample means and obtain the interval [2.5, 6.3]. What can you conclude?

• There is likely no difference between the population means

• The confidence interval is not valid.
• The population mean of group A is likely higher than that of group B by an amount between 2.5 and 6.3.

• The population mean of group B is likely higher than that of group A by an amount between 2.5 and 6.3.

If a confidence interval for the difference between two means is [−3.2,0.5], what does this imply about the population means?

• he population means are likely equal

• Population B's mean is likely higher than Population A's mean

• Population A's mean is likely higher than Population B's mean

• Since the confidence interval includes 0, we cannot rule out the possibility that there is no difference between the population means.

Which of the following statements about the Student's t-distribution is true?

• It is used when sample sizes are large and variances are known.

• It is identical to the normal distribution regardless of sample size.

• It has thicker tails than the normal distribution and used when the population standard deviation is unknown

• The Student's t-distribution is used when the population mean is unknown

What is the relationship between confidence level and the width of a confidence interval?

• As the confidence level increases, the width of the interval decreases.

• As the confidence level increases, the width of the interval increases.

• There is no relationship between confidence level and interval width.

• The width of the interval is always fixed regardless of the confidence level.