Weights of adult female California sea lions are approximately Normal with mean μ = 220 pounds and standard deviation σ = 8 pounds. What is the shape of the sampling distribution of x¯ created from random samples of size 36?
NormalSince the shape of the population is approximately Normal, the shape of the sampling distribution will be Normal.
The shape of the sampling distribution of x¯ is also approximately normal. We can use the Central Limit Theorem to justify this. The Central Limit Theorem states that if a random sample of size n is taken from any population with mean μ and standard deviation σ, then the sampling distribution of x¯ will be approximately normal with mean μ and standard deviation σ/√n, as long as n is large enough (usually n ≥ 30).
In this case, we are taking random samples of size 36 from a population with mean μ = 220 pounds and standard deviation σ = 8 pounds. Therefore, the mean of the sampling distribution of x¯ will also be 220 pounds. The standard deviation of the sampling distribution of x¯ will be σ/√n = 8/√36 = 8/6 = 4/3 pounds.
Since the sample size is large enough (n = 36 ≥ 30), we can say that the shape of the sampling distribution of x¯ is approximately normal with mean μ = 220 pounds and standard deviation σ/√n = 4/3 pounds.
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