Determine the optimal decision tree for finding the (lower) median of six numbers a, b, c, d, e and f, with the additional assumption that among these numbers there are exactly three distinct numbers, each occurring exactly twice. In zoom lecture we denoted the height of the optimal decision tree by (2, 2, 2).

Create a data set with 8 points in it that has a mean of approximately 10 and a standard deviation of approximately 1.

Atextile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. the company wishes to test the hypothesis upper h subscript 0 baseline colon mu equals 12 against upper h subscript 1 baseline colon mu less-than 12 using a random sample of n equals 4 specimens. calculate the p-value if the observed statistic is x overbar equals 11.8. round your final answer to five decimal places (e. g. 98.76543).

What is the value of the discriminant of the quadratic equation -2x = -8x + 8 and what does its value mean about thenumber of real number solutions the equation has?