Probstat/homework 3

This is part of probstat.

Part 1: Poisson and exponential random variables

1. Consider the number of car accidents in the parking lot over a period of one week. We know that the numbers of accidents are Poisson distributed random variable, and on average, there are 0.4 accidents per week. Find the probability that at least one accident occurs this week.

2. On average, each day Programmer Lazy puts in 5 bugs to the code. The manager tries to give incentives for the programmer to improve, so the manager decides to give a reward to the programmer on any day Programmer Lazy only puts in less than 3 bugs. If the number of bugs is a Poisson distributed random variable and Programmer Lazy does not change at all, what is the probability that Programmer Lazy gets a reward on any day?

3. The time it takes for a web server to crash (in hours) is exponentially distributed with parameter $\lambda =0.001$. What is the probability that the web server crashes within 2 weeks (14 days) of operation?

4. A system consists of 5 components. It will fail if any of its components are broken. The lifetime of each component $i$ (in minutes) is exponentially distributed with parameter $\lambda _{i}$. What is the probability that the system works for $t$ minutes?

Part 2: Hypothesis testing

1. We are interested in testing if a normally-distributed population has mean zero. We know that the variance of the population is 17. If we decide to take a sample of size 15, design a hypothesis testing procedure with the level of significance $\alpha =5\%$. State the null hypothesis clearly and describe the criteria for accepting/rejecting the null hypothesis.

2. We are interested in testing if a normally-distributed population has mean at least 100. We know that the variance of the population is 30. If we decide to take a sample of size 10, design a hypothesis testing procedure with the level of significance $\alpha =5\%$. State the null hypothesis clearly and describe the criteria for accepting/rejecting the null hypothesis.

3. We are interested in testing if a normally-distributed population has mean at least 100. However, we do not know the population variance. If we decide to take a sample of size 10 to compute the sample mean ${\bar {X}}$ and the sample variance $S^{2}$, design a hypothesis testing procedure with the level of significance $\alpha =10\%$. State the null hypothesis clearly and describe the criteria for accepting/rejecting the null hypothesis.

4. We want to test if students who have breakfast perform better in the exam than students who do not. From the data we collect over time, we believe that test scores are normally distributed and the variance of the test score is 120. We sample 30 students from the first group and 30 students from the second group, and collect their test scores. Describe a null hypothesis and show how to perform a hypothesis testing for this problem. Describe the criteria with 5% level of significance.

5. The time it takes for a web server to crash (in hours) is exponentially distributed. The company that builds the server claims that the server has an average up time of $h$ hours. We collect a sample of its up times $X_{1},X_{2},\ldots ,X_{20}$. Describe a hypothesis testing procedure with %5 level of significance that the average up time is at least $h$ hours as claimed by the company.

Part 3: Regression

1. Consider the following data:

x(i)	y(i)
73.88	22.67
62.66	27.48
94.99	45.46
84.45	34.61
46.05	28.39
75.46	34.32

• Estimate the parameter A & B such that $A+Bx$ is an estimate of $y$.

• Estimate the variance $\sigma ^{2}$ of the error.

2. (IPSES, problem 21, chapter 7.) Consider a different regression model where

$Y=\beta X+e,$

where $e\sim Normal(0,\sigma ^{2})$.

Suppose that we get a sample $(x_{1},y_{1}),\ldots ,(x_{n},y_{n})$. Find the least square estimator $B$ for $\beta$.