A population of values has a normal distribution with [tex]\mu = 99.8[/tex] and [tex]\sigma = 79.1[/tex]. You intend to draw a random sample of size [tex]n = 188[/tex].

a) What is the mean of the distribution of sample means? [tex]\mu_{\bar{x}} = [/tex]

b) What is the standard deviation of the distribution of sample means? (Report answer accurate to 2 decimal places.) [tex]\sigma_{\bar{x}} = [/tex]

b) The standard deviation of the distribution of sample means is equal to the population standard deviation divided by the square root of the sample size, σ/√n = 79.1/√188 ≈ 5.76.

a) The mean of the distribution of sample means is equal to the population mean. In this case, the population mean is given as μ = 99.8. When we draw multiple random samples from the population and calculate the mean of each sample, the average of all these sample means will be equal to the population mean.

b) The standard deviation of the distribution of sample means is determined by the population standard deviation and the sample size. In this case, the population standard deviation is given as σ = 79.1 and the sample size is n = 188. The standard deviation of the distribution of sample means is calculated by dividing the population standard deviation by the square root of the sample size, σ/√n. In this case, it becomes 79.1/√188, which is approximately equal to 5.76.

The standard deviation of the distribution of sample means represents the average amount of variation or dispersion among the sample means. As the sample size increases, the standard deviation of the distribution of sample means decreases, indicating a more precise estimation of the population mean.

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