4. In 2012 the Centers for Disease Control and Preventionreported that the rate of autism in children has risen to1 in 50. A study conducted in Norway suggests that therisk of autism may be significantly lower if women takefolic acid supplements during pregnancy. Suppose a largemedical center enlists 900 expectant mothers in a trial;the women agree to take the folic acid for the duration oftheir pregnancies. The researchers will track the childrenafter birth to see how many develop autism.a) Explain why the number of autism cases is a bino-mial random variable.b) If folic acid supplements actually have no effect, whatis the mean and standard deviation of the number ofautism cases that these researchers might find?c) How low would the number of autism cases amongthese 900 children have to be in order to convinceyou that the folic acid supplements may be effective?Explain your reasoning.5. An online retailer sells its $25 gift cards at supermarkets.The retailer knows that 20% of people who purchasethese cards spend the full value, 70% leave a balance of

Solution: From the given information, the rate of autism in children has risen to 1 in 50. The probability of autism in children is p=1/50. The sample size is n=900.a. The conditions for the binomial distribution are Trials are independent. Probability of success is same for all the trails. Number of trials should be fixed. There should be only two outcomes. That is success and failure. Let X be the number of autism cases. From the given information, number of expectant mothers is 900 and the probability of autism in children 1/50. Here, mothers are independent. The probability of success 1/50 is constant. Number of mothers is 900 and there are only two outcomes autism and no autism. The conditions for the binomial distribution are satisfied. Thus, the X follows binomial with parameters n=900 and p=1/50. Hence, the number of autism cases is a binomial random variable.b. The mean of the number of autism cases that these researchers is The standard deviation of the number of autism cases that these researchers is c. Since the mean number of autism cases is 18 the number of autism cases among these 900 children must be to convince that the folic acid supplements may be effective should be less than 18.