Question

Let x denote the time (in minutes) that it takes a fifth-grade student to read a certain passage. Suppose that the mean value and standard deviation of x are μ = 4 minutes and σ = 0.7 minutes, respectively.

(a)

If

x

is the sample mean time for a random sample of n = 9 students, where is the

x

distribution centered, and how much does it spread out around the center (as described by its standard deviation)? (Round your answers to three decimal places.)

μ _x

= minutes

σ _x

= minutes

(b)

Repeat part (a) for a sample of size of n = 20 and again for a sample of size n = 100. (Round your answers to three decimal places.)

n = 20

μ _x

= minutes

σ _x

= minutesn = 100

μ _x

= minutes

σ _x

= minutesHow do the centers and spreads of the three

x

distributions compare to one another?

The centers of the distributions of the sample mean are  ---Select--- larger for larger sample sizes smaller for larger sample sizes all the same as the population mean , and the standard deviations (and therefore spreads) of these distributions are  ---Select--- larger for larger sample sizes smaller for larger sample sizes all the same as the population standard deviation .

(c)

Which of the sample sizes in part (b) would be most likely to result in an

x

value close to μ, and why?A sample size of n =  ---Select--- 9 20 100 is most likely to result in a sample mean close to μ, since this is the sample size that results in the  ---Select--- smallest standard deviation largest standard deviation smallest mean largest mean of the distribution of

x.

Answer 1

a)

μ _x=µ=4.000 minutes

σ _x=σ/√n=0.7/√9 = 0.233 minutes

b)

n = 20

μ _x=4.000 minutes

σ _x=0.7/√20=0.157minutes

n = 100

μ _x=4.000 minutes

σ _x=0.7/√100=0.07 minutes

The centers of the distributions of the sample mean are  all the same as the population mean , and the standard deviations (and therefore spreads) of these distributions are   smaller for larger sample size

c)

A sample size of n=100 is most likely to result in a sample mean close to μ, since this is the sample size that results in the smallest standard deviation of the distribution of x.