Question

Suppose *x* has a distribution with μ = 60 and σ = 17.

(a) If random samples of size *n* = 16 are selected, can we say anything about the x distribution of sample means?

(b) If the original*x* distribution is *normal*, can we say anything about the x distribution of random samples of size 16?

(c) Find*P*(56 ≤ x ≤ 61). (Round your answer to four decimal places.)

No, the sample size is too small.

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 4.25.

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 17

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 1.1.

(b) If the original

No, the sample size is too small.

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 17.

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 1.1.

Yes, the x distribution is normal with mean μ_{ x} = 60 and σ_{ x} = 4.25.

(c) Find

Step 1

**Note:**

Hi there! Thank you for posting the question. The Part (b) of your question asks about the “x distribution of random samples of size 16”. However, on comparing it with the question in Part (a) that mentions “x distribution of sample means”, we have found it logical to assume that the question in Part (b) also enquires about the distribution of the sample means, and not just the distribution of x and solved the problem with that assumption.

Step 2

**Points on sampling distribution of the sample mean:**

- If the true population distribution of a random variable, say,
*x*, is normal with parameters, mean*μ*and standard deviation*σ*, then, whatever be the size (*n*) of the sample taken from the population, the distribution of the sample mean is also normal, with parameters, mean*μ*and standard deviation*σ/√n*. - Even if the true population distribution of a random variable, say,
*x*, is not normal and has population mean*μ*, standard deviation*σ*, then, for a large size (*n*≥ 30) of a sample taken from the population, the distribution of the sample mean is approximately normal, with parameters, mean*μ*and standard deviation*σ/√n*(by Central Limit Theorem). - If the true population distribution of a random variable, say,
*x*, is not normal and has population mean*μ*, standard deviation*σ*, then, for a small size (*n*< 30) of a sample taken from the population, the distribution of the sample mean cannot be said to be approximately normal.

Step 3

**(a)**

It is said that the distribution of *x* has population mean *μ* = 60 and population standard deviation *σ* = 17. The sample size, *n* = 16 is not large. Moreover, it is **not mentioned that the distributio...**

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