Final answer:
To find the probabilities and statistics for a binomial distribution with 8 shoppers and a probability of 0.2, you can use the binomial table and formula. For (a), look up the probabilities for each possible number of shoppers making a purchase and sum them. For (b), calculate the probabilities for 7 and 8 shoppers making a purchase separately and add them. For (c), use the formulas for mean and standard deviation.
Step-by-step explanation:
(a) To find the probability that among 8 shoppers, we can use the binomial table. The binomial table lists the probabilities for different values of n (the number of trials) and p (the probability of success). In this case, we have n = 8 and p = 0.2. We can look up the probability for each possible value of X (the number of shoppers making a purchase) in the table and add them up. In this case, we need to find the probabilities for X = 0, 1, 2, 3, 4, 5, 6, 7, and 8, and sum them.
(b) To find the probability that at least 7 customers will make a purchase using the binomial formula, we can calculate the probabilities for X = 7 and X = 8 (the probability of 0 shoppers making a purchase) separately, and add them up. The formula for the binomial probability is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.
(c) To find the mean and standard deviation, we can use the formulas for the mean and standard deviation of a binomial distribution. The mean of a binomial distribution is given by mean = n * p, and the standard deviation is given by standard deviation = sqrt(n * p * (1-p)). In this case, we have n = 8 and p = 0.2, so we can plug these values into the formulas to find the mean and standard deviation.