Question

Data on salaries in the public school system are published annually by a​ teachers' association. The mean annual salary of​ (public) classroom teachers is ​$53.2 thousand. Assume a standard deviation of ​$7.2 thousand. Complete parts​

(a) through​ (e) below. a. Determine the sampling distribution of the sample mean for samples of size 64. The mean of the sample mean is mu Subscript x overbarequals​$ nothing. ​(Type an integer or a decimal. Do not​ round.) The standard deviation of the sample mean is sigma Subscript x overbarequals​$ nothing. ​(Type an integer or a decimal. Do not​ round.) b. Determine the sampling distribution of the sample mean for samples of size 256. The mean of the sample mean is mu Subscript x overbarequals​$ nothing. ​(Type an integer or a decimal. Do not​ round.) The standard deviation of the sample mean is sigma Subscript x overbarequals​$ nothing. ​(Type an integer or a decimal. Do not​ round.) c. Do you need to assume that classroom teacher salaries are normally distributed to answer parts​ (a) and​ (b)? Explain your answer. A. ​No, because if x overbar is normally​ distributed, then x must be normally distributed. B. ​Yes, because the sample sizes are not sufficiently large so that x overbar will be approximately normally​ distributed, regardless of the distribution of x. C. ​No, because the sample sizes are sufficiently large so that x overbar will be approximately normally​ distributed, regardless of the distribution of x. D. ​Yes, because x overbar is only normally distributed if x is normally distributed. d. What is the probability that the sampling error made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 64 classroom teachers will be at most​ $1000? nothing ​(Round to three decimal places as​ needed.) e. What is the probability that the sampling error made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 256 classroom teachers will be at most​ $1000? nothing ​(Round to three decimal places as​ needed.) Click to select your answer(s).

Answer 1

Let X denote the salaries in the public school system.

a)

Using Central Limit Theorem, we have,

So,

b)

Using Central Limit Theorem, we have,

So,

c)

From Central Limit theorem, we know that when the sample size is sufficiently large, the sample means are Normally distributed irrespective of what the original distribution is.

So, no, we don't.

Hence option C is the correct option.

d) here

So,

Required probability =

=P(-1.11<Z<1.11)

=0.8667-0.1335

=0.7330

Anse

here

So,

=P(-2.22<Z<2.22)

=0.9869-0.01321

=0.9737