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A random sample is drawn from a population with mean μ=74 and standard deviation σ=6.2.

a. Is the sampling distribution of the sample mean with n=18 and n=47 normally distributed? (Round the standard error to 3 decimal places.)

User Kgilpin
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Final answer:

The sampling distribution of the sample mean for n=47 is normally distributed according to the Central Limit Theorem, but for n=18, it's only normal if the population is normally distributed. Standard errors for n=18 and n=47 are approximately 1.461 and 0.905, respectively.

Step-by-step explanation:

The question asks whether the sampling distribution of the sample mean for samples of size n=18 and n=47 is normally distributed given a population with mean μ=74 and standard deviation σ=6.2.

According to the Central Limit Theorem (CLT), the sampling distribution of the sample mean will be approximately normally distributed if the sample size is sufficiently large. The rule of thumb for a sufficiently large sample size is typically n ≥ 30. Therefore, for the sample size n=47, the sampling distribution of the sample mean is normally distributed.

However, for a smaller sample size such as n=18, the distribution will still be approximately normal only if the population from which the sample is drawn is itself normally distributed. Since this information is not provided, it's uncertain whether the sampling distribution for n=18 is normally distributed.

The standard error (SE) of the mean for each sample can be calculated using the formula SE = σ/√n, where σ is the population standard deviation and n is the sample size. For n=18, SE = 6.2/√18 ≈ 1.461 and for n=47, SE = 6.2/√47 ≈ 0.905. These values are rounded to three decimal places as requested.

User Tim Dams
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