Question

let ( x ) represent the full length of a certain species of newt. assume that ( x ) has a normal probability distribution with mean 64.5 inches and standard deviation 3.5 inches.
you intend to measure a random sample of ( n = 100 ) newts. the bell curve below represents the distibution of these sample means. the scale on the horizontal axis is the standard error of the sampling distribution. complete the indicated boxes, correct to two decimal places. use the exact value for your standard deviation in your calculations.

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Explanation:

Step1: Find the mean of the sampling distribution

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\).
Given \(\mu = 64.5\) inches, so \(\mu_{\bar{x}}=64.5\)

Step2: Find the standard deviation of the sampling distribution

The standard deviation of the sampling distribution of the sample mean (standard error) is given by the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)
Given \(\sigma = 3.5\) inches and \(n = 100\)

$$ \sigma_{\bar{x}}=\frac{3.5}{\sqrt{100}}=\frac{3.5}{10}=0.35 $$

Answer:

\(\mu_{\bar{x}} = 64.50\), \(\sigma_{\bar{x}}=0.35\)