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PLEASE ANSWER A regional bank conducted a study on the preferred payment method among American consumers. They found that in the general population 58% of individuals prefer using debit cards, 25% prefer using credit cards, and 17% prefer using cash. Now, the bank is interested in examining a random sample of 50 American consumers. (a) What is the sampling distribution for the sample proportion of individuals who have chosen debit card as their preferred payment method? (b) What is the sampling distribution for the sample proportion of individuals who have chosen credit card as their preferred payment method? (c) What is the probability that the sample proportion of individuals who have chosen credit card as their preferred payment method is higher than 35%? (d) What is the probability that the sample proportion of individuals who have chosen debit card as their preferred payment method is lower than 50%? Page 5 Homework 5 (e) What is the probability that sample proportion of individuals who have chosen debit card as their preferred payment method is between 0.6 and 0.7? (f) Comparing the sample proportion of those who have chosen debit card as their preferred payment method and those who have chosen credit card as their preferred payment method, which one has a lower standard deviation? Why do you think this is the case

To answer these questions, we can use the properties of the binomial distribution and the Central Limit Theorem. Let's calculate the required values step by step: (a) The sampling distribution for the sample proportion of individuals who prefer using debit cards can be approximated as a normal distribution. The mean of the sampling distribution is the same as the population proportion, which is 58%.

The standard deviation of the sampling distribution can be calculated as: ?_debit = sqrt((p_debit * (1 - p_debit)) / n), where p_debit is the population proportion (58%) and n is the sample size (50). (b) The sampling distribution for the sample proportion of individuals who prefer using credit cards can also be approximated as a normal distribution. The mean of the sampling distribution is the same as the population proportion, which is 25%.