Some of the math formatting is lost as HTML. If you can’t tell what is what, you can try a PostScript version which is available here: http://www.math.csi.cuny.edu/ verzani/410/test-2.ps
(I believe Adobe Acrobat will read this.)
This is test 2 for MTH410. It is a take home exam. This does not mean you can collaborate in anyway with your classmates or others. You are free to use outside sources such as books, the internet etc. But it is expected that you will not ask anyone else how to do these problems. If I suspect any violation of the above, I will schedule an oral examination.
That being said, good luck with this test. If you have difficulty with the wording of a question, feel free to contact me. In case you can’t come in, my e-mail address is verzani@math.csi.cuny.edu. Any changes to the test (I don’t expect any) will be announced on my homepage under the 410 link. (www.math.csi.cuny.edu/~verzani
You might want to use MATLAB or some other software to find some of the values.
Calculate $\bar X$ and $S$ for this data
$$7,9,12,31,12,8,2,45,5,10$$
Suppose $X$ has a chi-squared distribution with 10 degrees of freedom. Find $x$ so that $P(X > x) = 0.05$.
Suppose the following numbers are a random sample from a normal population with variance 2 and unknown mean. Find a 95% confidence interval for the mean.
4.1 6.9 5.8 1.8 1.7 4.9 6.1 4.5 1.5 6.7
An approximately normal population with unknown mean and variance is sampled randomly resulting in the numbers below. Find a 95% confidence interval for the mean.
4.1 6.9 5.8 1.8 1.7 4.9 6.1 4.5 1.5 6.7
A webmaster for statsRus.com is interested in estimating the variability in the day-to-day usage of her website. Suppose the number of visits in hundreds of thousands for one week is given by
4.1 6.9 5.8 1.8 1.7 4.9 6.1 If she assumes the population is approximately normal and the data is a random sample, find a 90% confidence interval for $\sigma^2$.
A random sample of size $n$ is taken from a population with Poisson distribution with parameter $\lambda$. Find the maximum likelihood estimator for $\lambda$.
A coin is to be used in the superbowl to determine who gets the opening kickoff. It is first tested to see if it is fair. Construct a statistical test to do this by specifying $H_0$, $H_A$ and a test statistic $Y$.
Toss a coin 20 times are write the results here. For this data set, perform your statistical test above at a level of $\alpha=0.10$. Do you reject or accept the null hypothesis? Why?
Find the likelihood ratio satistic
$$\Lambda = \frac{L(\hat \theta_0)}{L(\hat \theta)}$$
for a random sample of size $n$ from an exponential family with parameter $\lambda$ under the hypotheses
$H_0$: $\lambda = \lambda_0$
$H_A$: $\lambda \not = \lambda_0$
For these questions, please specify completely your solution including $H_0$, $H_A$ etc. as appropriate.
A giant soft drink company is testing a new cola drink that is lgiht, cheery and full of fun. Suppose it requires a 60% favorable rating for it to be successful. They employ statsRus.com to run a test to determine the favorability of the new drink. 100 people are surveyed at random and it is found that 62% find the new drink favorable. Is this statistical evidence to discard the null hypothesis for the alternative at the $\alpha=0.10$ level? Explain.
Ray has found a job with InsuranceRus.com and his first assignment is to determine if there has been insurance fraud by the autobody shops on Staten Island. He takes a random sample of 10 claims and finds that the average claim is $2542. Suppose, the claims are assumed to be normally distributed. Is the data sufficient evidence to reject the hypothesis that the mean claim is $2300? Explain.