You are interested in the probability of the outcome when rolling a six-sided die. Answer the following questions using p to denote the probability of this outcome.

(a) You suspect that for this die the outcome does not occur with the same probability as with a fair die. Formulate the null and alternative hypotheses based on which you may make a statistical inference regarding your suspicion.

(b) You suspect that for this die the outcome occurs with greater probability than with a fair die. Formulate the null and alternative hypotheses based on which you may make a statistical inference regarding your suspicion.

(c) You suspect that for this die the outcome occurs with lesser probability than with a fair die. Formulate the null and alternative hypotheses based on which you may make a statistical inference regarding your suspicion.

(d) You believe that for this die the outcome occurs with the same probability as with a fair die.

How might you collect evidence in support of your belief?

Which is easier: to collect evidence in favor of balancedness or to collect evidence against b

(10) 4. Answer the following: (5) a) Differentiate between descriptive and inferential statistics with an example for each. (5) b ) State the Central Limit Theorem (CLT). The formal statement is the most important theorem of probability and concept of statistical inference. Why? Explain your answer using graphs to illustrate your point.1. What is the physical interpretation of (x, t) for any complex quantum mechan- ical wave function V(x, ()? 2. A Fourier transform may be used to transform the Gaussian wave packet from po- sition space representation I(r, t) into momentum space (k-space) representation V ( k, t). What function is the Fourier transform of a Gaussian function? What is the physical interpretation of | V(k, t) |2? 3. Let us define Ar and Ak to be the widths of a Gaussian wave packet in position space representation V(r, f) and in momentum space representation V(k, t), re- spectively. What is the specific property of a Gaussian wave packet regarding its position-momentum uncertainty ArAk? 4. Why can the transfer matrix method not be used to examine the time evolution of a Gaussian wave packet?28. Making decisions based on probability involves: a) Descriptive Statistics b) Statistical Inference c) Conditional Probability d) Response bias 29. Which of the answers below are methods of Power transformation? a) Square root, Cube root, Logarithm b) Median, Mode, Mean c) Slope, y-intercept d) None of the above