Question

Let x be a random variable representing the amount of sleep each adult in New York City got last night. Consider a sampling distribution of sample means x. (a) As the sample size becomes increasingly large, what distribution does the x distribution approach? uniform distribution sampling distribution normal distribution binomial distribution (b) As the sample size becomes increasingly large, what value will the mean μx of the x distribution approach? μx μ μ/√n μ/n σ (c) What value will the standard deviation σx of the sampling distribution approach? σ/n σx σ/√n μ σ (d) How do the two x distributions for sample size n = 50 and n = 100 compare? (Select all that apply.) The standard deviations are μ / 50 and μ / 100, respectively. The means are the same. The standard deviations are the same. The standard deviations are σ / √50 and σ / √100, respectively. The standard deviations are μ / √50 and μ / √100, respectively. The standard deviations are σ / 50 and σ / 100, respectively.

Answer 1

a) As the sample size becomes increasingly large, the distribution approximates a normal distribution.

b) As the sample size becomes increasingly large, the mean of the sampling distribution, μₓ, approach as the population mean, μ.

σₓ

c) The standard deviation of the sampling distribution is given as

σₓ = (σ/√n)

d) Two sampling distributions with Sample size 50 and 100 have standard deviation of sampling distribution of (σ/√50) and (σ/√100) respectively.

Explanation:

The Central limit theorem explains that sample distributions obtained from a population distribution (especially normally distributed ones) have distributions that approximate a normal distribution, a mean that is equal to the population mean and a standard deviation of sampling distribution given as the population standard deviation divided by square root of sample size.

a) Just like the Central limit theorem explains, as the sample size becomes increasingly large, the distribution approximates a normal distribution

b) And as the sample size becomes increasingly large, the mean of the sampling distribution, μₓ, approach as the population mean, μ.

c) The standard deviation of the sampling distribution is given as the population standard deviation divided by square root of sample size.

d) Using the formula given in (c) above, the standard deviation of sampling distribution of two sampling distributions with sample size 50 and 100 are (σ/√50) and (σ/√100) respectively.

Hope this Helps!!!

Answer 2

Explanation:

Given that X is a random variable representing the amount of sleep each adult in New York City got last night.

a) Normal distribution (by central limit theorem)

b) Sample mean will approach population mean mu

c) the standard deviation σx of the sampling distribution approach to σ/√n

d) The means are the same.

. The standard deviations are σ / √50 and σ / √100, respectively.