MAT 2375
Practice Exam
1. [5 points] Let X1, . . . , X25 be a random sample from a N(µ = 10, σ2 = 9) population. Let and S be the sample mean and sample standard deviation, respectively.
(a) Determine the following probability
(b)
2. [5 points] Let X equal the number of pounds of butterfat produced by a Holstein cow during the 305-day milking period following the birth of a calf. Assume a normal population. We would like to test H0 : µ = 715 against H1 : µ < 715 at a level of significance of α = 5%.
From a random sample of 25 cows, we observed = 708 and s2 = 245.
(a) Compute the observed value of the test statistic.
(b) Give the p-value.
(c) Give the conclusion at α = 5%.
3. [5 points] Let X1, . . . , Xn be a random sample from an Poisson population with mean θ = λ, that is from a population with the probability mass function
(a) Determine the likelihood function L(θ).
(b) Determine the maximum likelihood estimator for θ.
(c) Is the maximum likelihood estimator of θ unbiased? Justify.
4. [5 points] A production process for making bearings with mean diameter µ = 0.5 centimeters (cm) is under investigation. An engineer collects a random sample of 10 bearings. The observed sample mean is = 0.506 cm and sample standard deviation is s = 0.004 cm.
(a) A 95% confidence interval for the mean diameter µ.
(b) If we want the maximum error of the estimate to be less than 0.001 cm, with a level of confidence of 95%, how large should the sample size n be? Assume that the population standard deviation is σ = 0.004 cm.
5. [5 points] The time taken in minutes to complete a particular task in a widget factory follows a normal distribution with mean 30 and variance 1. In an effort to reduce the mean time required for the task, 16 workers are asked to carry out the task using a new procedure. The procedure will be adopted that if H0 : µ = 30 is rejected and H1 : µ < 30 is accepted at a significance level of α = 5%. Assume that the population is still normal and that the variance remains unchanged for this test procedure, that is the population variance is 1.
(a) Determine the power of the test, if in fact µ = 29, that is determine K(29).
(b) We would like the power of the test to be 80% if µ = 29 at a level of significance of 5%. Give the required sample size n.
6. [5 points] We wish to compare two treatments. Treatment 1 is given to 22 subjects selected at random, and treatment 2 is given to 15 other subjects selected at random. Assume independent normal populations with equal variances.
The data yielded:
We would like to test H0 : µX − µY = 0 against H1 : µX − µY ≠ 0.
(a) Give the observed value of the test statistic, the p-value, and the conclusion at α = 5%.
(b) Give a 95% confidence interval for µX − µY .
7. [5 points] Although the silicon dissolved in sea water is not required by all primary producers, studies have linked the depletion of the substance with decreased productivity. A study is being conducted to better understand the behaviour of dissolved silicon. The two studied variables are x = distance in kilometers from shore, and y = concentration in micrograms per silicon litre. Here are the data
The data may be summarized as follows:
Assume that is it reasonable to assume
Yi = α + β (xi − ) + εi
,
where ε1, . . . , εn is a random sample from a N(0, σ2
) population.
(a) Give the least-squares line that expresses the concentration according to distance from the shore.
(b) Give an estimate of the variance of the random error, that is give ˆσ
2
.
(c) We want to test H0 : β = 0 against H1 : β ≠ 0 at α = 5%. Give the observed value of the test statistic, the p-value, and the conclusion.