An electronic system has four components labeled as 1, 2, 3, and 4. The system has to be used during a given time period. The probability that component i will fail during that time period is fi for i = 1, . . . , 4. Failures of the components are physically independent of each other. A system failure occurs if component 1 fails or if at least two of the other components fail. Specify an appropriate sample space and determine the probability of a system failure.

(Jeff Bezos), Bill Gates (BG) 66 Elon Musk (EM) 50 Jeff Bezos (JB) 57 Mark Zackenberg (MZ) 37 Warren Buffet (WB) 91 1. Assuming that 2 of the ages are randomly selected with replacement, list the 25 possible samples 2. Find the mean of each of the 25 samples, then summarizing the sampling distribution of the means in the format of a table representing the probability distribution. 3. Compare the population mean to the mean of the sample means. 4. Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? 9

whether the residents' satisfaction with the available services depends on their gender. A random sample of A rural county hospital offers several health services. The hospital administrators conducted a poll to determine whether he or she was satisfied with the services offered by the hospital. The resulting data are shown in the 1,000 adult county residents was selected. The gender of each respondent was recorded and each was asked table below. Satisfied Not Satisfied Total Male 384 b) Find the expected counts for each cell. 80 464 a) What are the null and alternative hypotheses? c) Compute the Chi-Squared test statistic. Is there an association between gender and satisfaction with health services? Female 416 120 536 Total 800 200 1,000 d) Given that the p-value is 0.042 what is your decision in the context of the question if alpha= .05? W

Suppose we have a sample with five equally likely experimental outcomes: E1, E2, E3, E4, E5 Let: A = (E1, E3, E4, E5) B= (E1, E3} C = (E2, E3, E4,) a) There are 3 methods of assigning probabilities - classical, subjective and relative frequency. Which method is used above? b) Find P(A), P(B) and P(C) c) Find A and P(AG) d) Find P(B U C^C) e) Are A and C mutually exclusive? Explain.