A variable of a population has a mean of 32 and a standard deviation of 12. (a)

Question

A variable of a population has a mean of 32 and a standard deviation of 12. (a) If the distribution of the variable is unknown, identify the sampling distribution of the sample mean for samples of size 64. (b) Can you answer part (a) if the distribution of the variable under consideration is unknown and the sample size is 34 instead of 64? Explain your answer. (c) Can you answer part (a) if the distribution of the variable under consideration is unknown but the sample size is 18 instead of 64? Why or why not?

Explanation / Answer

Mean=32, Standard deviation = 12

Solution (a) According to Central limit theorem, given a population with a finite mean and a finite variance 2, the sampling distribution of the mean approaches a normal distribution with mean and a variance of 2/N as N, the sample size, increases.

Hence, mean of sampling distribution = 32

Variance of sampling distribution = 122/64 = 2.25

So, the sampling distribution of sample mean approches a Normal distribution with mean 32 and variance 2.25

Solution (b)   If the sample size 34 instead of 64,

Mean of sampling distribution = 32

Variance of sampling distribution = 122/34 = 4.24

So, the sampling distribution of sample mean approches a Normal distribution with mean 32 and variance 4.24.

(A sample size of 30 is sufficiently large enough for the sampling distribution of the mean to look approximately Normal)

Solution (c) Since, the population has a finite mean and a finite non-zero variance, the population is normal. (Again refer to the statement of the Central limit theorem provided in Solution (a)). Hence, for sample size = 18,

Mean of sampling distribution = 32

Variance of sampling distribution = 122/18 = 8

So, the sampling distribution of sample mean approches a Normal distribution with mean 32 and variance 8.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.