If x has a normal distribution with mean μ = 15 and standard deviation σ = 3, describe the distribution of x ¯ values for sample size n where n = 4, n = 16 and n = 100. How do the x ¯ distributions compare for the various sample sizes?

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If x has a normal distribution with mean   μ  =  15 and standard deviation   σ  =  3, describe the distribution of       x    ¯   values for sample size n where n = 4, n = 16 and n = 100. How do the       x    ¯   distributions compare for the various sample sizes?

adwelirlz · Accepted Answer

Given:  μ  =  15  σ  =  3      n          1        =  4      n          2        =  16      n          3        =  100Central limit theorem: If the sample size is large (30 or more), then the sampling distribution of the sample mean       x    ¯   is approximately normal with mean   μ and standard deviation       σ          n      The two       x    ¯   distributions for sample size n=50 and n=100 will all be a normal distribution (since the population distribution is normal) and will have the same mean   μ  =  15.The standard deviation will be different for each sample:      n          1        =  4  ⇒      σ                  x        ¯              =      σ          n        =      3          4        =  1.5      n          2        =  16  ⇒      σ                  x        ¯              =      σ          n        =      3          16        =  0.75      n          3        =  100  ⇒      σ                  x        ¯              =      σ          n        =      3          100        =  0.3Result:Both have approximately a normal distribution.Both have the same mean.Standard deviations are different for each sample (1.5, 0.75, 0.3).

If x has a normal distribution with mean mu = 15 and standard deviation sigma = 3, describe the distribution of bar(x) values for sample size n where n = 4, n = 16 and n = 100. How do the bar(x) distributions compare for the various sample sizes?

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