Please help with the following questions

1. Data collected over a long period of time showed that a particular genetic defect occurs in 2 of every 1000 children. Let X be the random variable "number of children with genetic defect in a sample of 20,000 children examined". The records of a medical clinic show X = 25 children.

a. What is the probability of observing a value of X more than 25?

b. Would you say that the observation of X = 25 children with genetic defect was rather unlikely?

2. Suppose a random sample of n = 15 observations is selected from a population that has normal distribution with mean 50 and standard deviation 10.

a. Give the mean and the standard deviation of the sample mean ???? .

b. Find the probability that ???? exceeds 55.

c. Find the probability that the sample mean deviates from the population mean by less than 4.

3. By statistics, faculty with rank of assistant professor (AP) finishing their 2nd year of employment at a higher education institution in Ontario earn an average of $ 75,360 per year with a standard deviation of $2,800. In an attempt to verify this salary level, a random sample of 40 AP with 2 years of experiment was selected from a personnel database for all higher education institutions in Ontario.

a. Describe the sampling distribution of the sample mean ???? of the average of these 40 AP.

b. Within what limit would you expect the sample mean to fall with probability .95?

c. If the random sample actually produced a sample mean ???? = 72, 000, would you consider this rather unusual? What conclusion might you draw then?

4. In a report on why e-shoppers abandon their online sale transactions, a study found that "pages took me too much time to load" and "site was too confusing to me so that I couldn't find the product" were the two main complaints heard most often. Based on 50 customers' responses, the average time to complete online order was 5.3 minutes and the standard deviation was 3.1 minutes. Construct a 95% confidence interval for the average completion time for an online order.

5.In a poll of n = 700 randomly selected adults, 390 indicated that movies are getting better. Construct a 90% confidence interval for the overall proportion ???? of adults who say that movies are getting better.

6. Suppose the number of successes in n = 200 binomial trials is 25.

a. Find an 80% confidence interval (CI) for ????, the probability of success at each trial.

b. Find a 90% CI for p. Why is this interval wider than the previous one?

7. (choosing sample size). Suppose you wish to estimate the mean pH of rainfalls in an area that suffers heavy pollution due to the discharge of smoke from a power plant. Previous studies showed that the standard deviation is in the neighborhood of .6 pH, and you wish your estimate to lie within .1 of the unknown mean , with probability .80. Approximately how many rainfalls must be included in your sample?