Question

complete parts (a) through (d) for the sampling distribution of the sample mean shown in the accompanying graph
click the icon to view the graph.
(a) what is the value of \\( \mu _ { \overline { x } } \\)?
the value of \\( \mu _ { \overline { x } } \\) is 400
(b) what is the value of \\( \sigma _ { \overline { x } } \\)?
the value of \\( \sigma _ { \overline { x } } \\) is
sampling distribution
the distribution is normal. the locations of the
peak and inflection points are shown.

Explanation:

Step1: Recall the property of normal distribution

For a normal distribution of the sample mean \(\bar{X}\), the standard deviation of the sample mean \(\sigma_{\bar{X}}\) can be found using the relationship between the inflection points. The distance between the mean \(\mu_{\bar{X}}\) and an inflection point is equal to \(\sigma_{\bar{X}}\).

Step2: Calculate \(\sigma_{\bar{X}}\)

We know that \(\mu_{\bar{X}} = 400\). One of the inflection points is at \(x = 370\). Then \(\sigma_{\bar{X}}=\mu_{\bar{X}}- 370\) (or \(430-\mu_{\bar{X}}\)).

$$ \sigma_{\bar{X}}=400 - 370=30 $$

Answer:

The value of \(\sigma_{\bar{x}}\) is \(30\)