A simple random sample of size nequals=45 is obtained from a population with mue

Question

A simple random sample of size

nequals=45

is obtained from a population with

muequals=64

and

sigmaequals=16

(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of

x overbarx.

(b) Assuming the normal model can be used, determine

P(x overbarxless than<67.5).

(c) Assuming the normal model can be used, determine

P(x overbarxgreater than or equals65.6).

(a) What must be true regarding the distribution of the population?

A.

The population must be normally distributed and the sample size must be large.

B.

The sampling distribution must be assumed to be normal.

C.

The population must be normally distributed.The population must be normally distributed.

D.

Since the sample size is large enough comma the population distribution does notSince the sample size is large enough, the population distribution does not

need to be normal.need to be normal.

Explanation / Answer

(a) Since sample size is greater than 45 and sample mean is to be determined, the correct answer as per Central limit theorem is

D.

Since the sample size is large enough, the population distribution does not need to be normal.need to be normal.

(b) P(x' < 67.5)

= P(z < ((67.5 - 64) / 16 * 45))

= P(z < 1.4674)

From tables, the value is 0.9289.

(c) P(x' >= 65.6)

= P(z >= ((65.6 - 64) / 16 * 45))

= P(z >= 0.6708)

From tables, the value is 0.2512.

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