A random sample of size n = 79 is taken from a finite population of size N = 674

Question

A random sample of size n = 79 is taken from a finite population of size N = 674 with mean = 253 and variance 2 = 434. Use Table 1.

Is it necessary to apply the finite population correction factor?

Calculate the expected value and the standard error of the sample mean. (Round “expected value” to a whole number and "standard error" to 4 decimal places.)

What is the probability that the sample mean is less than 241? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

What is the probability that the sample mean lies between 247 and 262? (Use rounded standard deviation. Round "z" value to 2 decimal places and final answer to 4 decimal places.)

A random sample of size n = 79 is taken from a finite population of size N = 674 with mean = 253 and variance 2 = 434. Use Table 1.

Explanation / Answer

Part a.1

Yes, it is necessary to apply the finite population correction factor; because sample size is more than 5% of population size.

Population size = N = 674

Sample size = n = 79

Sample size n = 79 > 5% of population size 674 = 33.7

Part a.2

We know that the expected value is same as the population mean µ.

So,

Expected value = µ = 253

Now, we have to find standard error.

Formula for standard error with population correction factor is given as below:

Standard error = [/sqrt(n)]*sqrt[(N – n)/(N – 1)]

We are given

^2 = 434,

So = sqrt(434) = 20.83267

N = 674

n = 79

Standard error = [20.83267/sqrt(79)]*sqrt[(674 – 79)/(674 – 1)]

Standard error = 2.343858* 0.940266

Standard error = 2.2039

Part b

We have to find P(Xbar < 241)

Z = (Xbar - µ) / Standard error

Z = (241 – 253) / 2.2039

Z = -5.44

P(Z<-5.44489) = P(Xbar < 241) = 0.0000000266

P(Xbar < 241) = 0.0000 approximately

Part c

Here, we have to find P(247<Xbar<262)

P(247<Xbar<262) = P(Xbar<262) – P(Xbar<247)

First we have to find P(Xbar<262)

Z = (Xbar - µ) / Standard error

Z = (262 - 253) / 2.2039

Z = 4.08

P(Z<4.08) = 0.999977482

P(Xbar<262) = 0.999977482

Now, we have to find P(Xbar<247)

Z = (247 - 253) / 2.2039

Z = -2.72

P(Z<-2.72) = P(Xbar<247) = 0.003264096

P(247<Xbar<262) = P(Xbar<262) – P(Xbar<247)

P(247<Xbar<262) = 0.999977482 - 0.003264096

P(247<Xbar<262) = 0.996713386

Required probability = 0.9967

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