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A and B are two events with positive probabilities. Define the following statements:

In testing a statistical hypothesis,  the p-value of the test statistic based on a random sample is 0.045.      Determine whether the following statement is correct:                  If  the significance level (or type I error) is 0.05,  we reject the null hypothesis.

A random sample of 50 observations produced a mean value of 55 and standard deviation of 6.25.  What is the 95% confidence interval for the population mean?   

The following observations are the numbers of customers in a service center in ten days.          9, 13, 8, 14, 18, 6, 11, 12, 10, 7 Find the range.

Which of the following statements is always true? A.  The union of two events A and B is the event consisting of all outcomes that are in both events. B.  The intersection of two events A and B is the event consisting of all outcomes that are either in A or in B. C.  The union of events A and B contains more sample points than their intersection. D.  The complement of an event A is the set of all outcomes in the sample space that are not contained in A.

Two independent samples of sizes  are randomly selected from two normal populations with unknow variances,  Assume two variances are equal.  The sample means and the sample variances  are as follows:  

Two discrete random variables X and Y have the joint probability function f(x,y): f(x,y) x=0 1 2 y=0 0.10 0.20 0.10 2 0 0.10 0.10 4 0.20 0 0.20               Find the covariance of X and Y:   Cov(X,Y).

A random variable X has a density function . Find the expectation  E(X2). 

A rare disease occurs with probability 0.002 in a large population.  Now we select 1000 people at random.  Find the probability that at most 3 people have contracted the disease.  

Extensive experience with fans of certain type used in diesel engines has suggested that the exponential distribution provides a good model for time until failure. Suppose the mean time until failure is 20,000 hours. What is the probability that a randomly selected fan has a lifetime less than 10,000 hours?