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 normal distribution binomial distribution sampling distribution

(b) As the sample size becomes increasingly large, what value will the mean μx of the x distribution approach?

μ/√n μx σ μ μ/n

(c) What value will the standard deviation σx of the sampling distribution approach?

σ/n σ/√n σx σ μ

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

The standard deviations are the same.

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

The means are the same.

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

Answer 1

(a) As the sample size becomes increasingly large, what distribution does the x distribution approach?

As n increases the distribution tends to normal

normal distribution

(b) As the sample size becomes increasingly large, what value will the mean μx of the x distribution approach?

As n increases, sample mean approaches population mean

μx = μ

(c) What value will the standard deviation σx of the sampling distribution approach?

As n increases, the standard deviation reduces

σx =  σ/√n

(d) How do the two x distributions for sample size n = 50 and n = 100 compare?

Mean will be same only standard deviation will change

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

The means are the same.