In earlier class units, we saw that a dataset that included the mean values derived from sampling from a dataset with an unknown distribution could be treated as a new random variable, and that this new variable had several important properties. Explain how collecting mean values (X-bar) from samples drawn from a dataset, regardless of the statistical distribution of the dataset, results in a distribution of those values that is centered on the mean value of the original dataset and is approximately normally distributed depending upon the sample size used. Include the name the theorem that describes this phenomenon, and discuss what it means to treat the X-bar data points derived from sampling as a new random variable. Why is this important in engineering

Nthe opening scene of the movie raiders of the lost ark, indiana jones replaces a golden statue with a bag of sand. the platform on which the statue is placed is designed to detect the mass of the statue. if the bag of sand has a different mass than the statue, then a mechanism triggers and destruction ensues. the density of gold is about 20\text{ g/cm}^320 g/cm 3 20, space, g, slash, c, m, start superscript, 3, end superscript. the density of sand can be estimated at 2.5\text{ g/cm}^32.5 g/cm 3 2, point, 5, space, g, slash, c, m, start superscript, 3, end superscript. the gold statue can be modeled as a cylinder with height 25\text{ cm}25 cm25, space, c, m and radius 8\text{ cm}8 cm8, space, c, m. in terms of \piπpi, what is the mass of sand indiana jones needs to avoid triggering the mechanism?