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In a small island, earthquakes occur twice per year on average. Assume the number of the earthquakes follows from a Poisson distribution. What is the probability that there are 3 to 6 earthquakes from January 1,  2029 to  December 31, 2030?

Assume α=3, β=λ=2.  The lifetime (in years) for a certain type of batteries has a gamma distribution with  α and  β.  Randomly select 100 batteries. What is the expected number of batteries whose lifetimes are at most  λ years?

A normal population has a standard deviation

A candidate for public office is thought to have support of 60% of the voters. Randomly select 100 voters.  Use normal approximation to find the probability for the event that between 50 and 60 (exclusive) voters will support the candidate.

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: