Probability #3 - Questions and Answers

Question 1 A mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time. If the company really ships 90% of its orders on time, what is the probability that the proportion in an SRS of 100 orders is 0.86 or less? Answer Given: p=90%=0.90 x=86 n=100 The sample proportion is calculated by dividing the number of successes by the sample size: p̂ = x/n = 86/100 = 0.86 The mean of the sampling distribution of p̂ is equal to the population proportion p: μp̂ = p = 0.90 The sampling distribution standard deviation of p̂ is: σp̂ = √(p(1-p)/n) = √(0.90(1-0.90)/100) = √0.0009 = 0.03 The z-score is the value decreased by the mean, divided by the standard deviation: z = (x - μ) / σ = (0.86 - 0.90) / 0.03 = -0.04 / 0.03 = -1.33 Using table A, calculate the corresponding probability: P(p̂ ≤ 0.86) = P(z < - 1.33) = 0.0918 Question 2 Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X? Answer Let X represent the difference between the number of heads and the number of tail obtained when a coin is tossed n times. We are interested in all possible outcomes of X. In n coins tossed tails can be obtained 0,1,2,...,n times. If tails are obtained k times number of heads or obtained precisely n-k times. Therefore, let us t be the number of tails and h=n-t number of heads. Lets define X = h - t = n - t - t = n - 2t Therefore all possible outcomes of X can be described by {n - 2t | t ∈ {0,1,...,n}}. Thus, we done. Result: Possible values of X are n - 2t, where t = 0,1,...n. Question 3 Classify the statement as an example of classical probability, empirical probability, or subjective probability. The probability that a newborn baby is a boy is 1/2. -empirical -classical -subjective Answer The statement is "The probability that a new-born baby is a boy is 1/2". Classical Probability: If each outcome in a sample space is equally likely to occur then it is said to have a uniform probability distribution. Empirical Probability: The empirical probability is based on observations obtained from probability experiments. The empirical probability of an event is the relative frequency of the event. Subjective Probability: If the probabilities result from intuition, educated guesses and estimates then the probability is said to be the subjective probability. Here, it is probable that the statement is based on empirical evidence. Thus, the statement "The probability that a new-born baby is a boy is 1/2" is an example of the empirical probability". Question 4 If you are aware that two events, A and B, are independent P(A)=0.3, what is the value of P(A|B)? Answer If A and B are independent, the likelihood of A happening is not dependent on event B occurring. Hence P(A|B)=P(A). Since P(A)=0.3, then P(A|B)=0.3. Question 5 A company with a fleet of 150 vehicles discovered that seven out of the 22 emissions systems they evaluated did not adhere to pollution control standards. Is this conclusive proof that up to 20% of the fleet may not be in compliance? Test a relevant theory, then provide your findings. Before moving on, confirm that all necessary presumptions and criteria have been met. Answer Ho: p = 0.20 Ha: p > 0.20 Assumed SRS. sample is less than 10% of population. (0.20)(22) < 10 and (0.80)(22) > 10