In New York City, a study was conducted to evaluate whether any information that is available at the time of birth can be used to identify children with special educational needs. In a random sample of 45 third-graders enrolled in the special education program of the public school system, 4 have mothers who have had more than 12 years of schooling. (a) Construct a 90% confidence interval for the population proportion of children with special educational needs whose mothers have had more than 12 years of schooling. (b) In 1980, 22% of all third-graders enrolled in the New York City public school system had mothers who had had more than 12 years of schooling. Suppose you wish to know whether this proportion is the same for children in the special education program. What are the null and alternative hypotheses of the appropriate test? (c) Conduct the test at the 0.05 level of significance. (d) What do you conclude? (e) If the true population proportion of children with special educational needs whose mothers have had more than 12 years of schooling is as low as 0.10, you want to risk only a 5% chance of failing to reject the null hypothesis. If you are conducting a two-sided test at the 0.05 level of significance, how large a sample would be required?