[1] 1 1 1 2 4 2 2 1 2 4 3 1 1 2 1 4 1 4 3 4 4 2 3 1 1 2 2 1 2 2Practice exercises for the final exam
Exercise 01
Suppose you are in the situation where you have independent samples from two normal populations: \[X_{1},\ldots,X_{n}\sim N\left(\mu_{1},\sigma^{2}\right),\ Y_{1},\ldots,Y_{n}\sim N\left(\mu_{2},\sigma^{2}\right),\] where \(\mu_{1},\mu_{2},\sigma^{2}\) are unknown.
Consider the following three estimators of \(\sigma^2\): \[S^2_X=\frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}\right)^2, \ S^2_Y=\frac{1}{n-1}\sum_{i=1}^n\left(Y_i-\overline{Y}\right)^2, \ S^2=\frac{1}{2}\left(S^2_X+S^2_Y\right)\]
- Are these estimators unbiased for \(\sigma^2\)?
- Compute the variances of each of these estimators. Which has the smallest variance?
- Explain why a pooled variance as a plug-in for \(\sigma^2\) is used in forming the test statistic when testing the null that \(\mu_1=\mu_2\) when \(\sigma^2\) is unknown.
Exercise 02
Let \(X_{1},\ldots,X_{16}\) be IID \(N\left(50,100\right)\). Your task is to express \[\mathbb{P}\left(796.2<{\displaystyle \sum_{i=1}^{16}}\left(X_{i}-50\right)^{2}<2630\right)\] in the form \(\mathbb{P}\left(c_{1}\leq V\leq c_{2}\right)\) where \(V\) is a random variable which has a known distribution. You have to provide the forms of \(c_{1},c_{2},V\) and the distribution of \(V\). If possible, find the exact value for the desired probability. If it is not possible to do this, provide a range.
Exercise 03
Let \(X_{1},\ldots,X_{5}\) be IID \(N\left(0,4\right)\). Let \[V=\dfrac{\left(X_{1}-X_{2}\right)}{\sqrt{X_{3}^{2}+X_{4}^{2}+X_{5}^{2}}}\] Find the constant \(c\) so that \(cV\) will have a \(t\)-distribution.
Exercise 04
Some games use four-sided dice for gameplay. These dice are designed to produce equally likely outcomes. Suppose you have a four-sided die and you roll this die independently \(n\) times. Suppose we observed 30 rolls of this die:
The sample mean is 2.13 while the sample standard deviation is 1.14.
Given the context, what hypotheses are you testing? What is the meaning of the parameters in this context? Conduct a test to determine whether, on the basis of 30 rolls of your die, your die is producing equally likely outcomes or your die is not producing equally likely outcomes.
Exercise 05
(Based on Devore et al 2022) Suppose you are provided the following two-way table of sample proportions in various combinations of categories.
| 1 | 2 | 3 | |
| 1 | 0.13 | 0.19 | 0.28 |
| 2 | 0.07 | 0.11 |
What is the smallest sample size \(n\) for which these observed proportions would result in rejection of the null hypothesis of independence at the 10% level?
Exercise 06
(Based on Devore et al 2022) In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125-cc two-stroke motorcycle, five plugs of each brand were tested for the number of miles (at a constant speed) until failure. A client asked you to conduct an analysis based on the partial ANOVA table for the data is given below.
| Source | Sum of squares | df | Mean sum of squares | F |
|---|---|---|---|---|
| Brand | ||||
| Error | 14713.69 | |||
| Total | 310500.76 | |||
- What do you think were the hypotheses being tested? Describe what are the parameters and assumptions in this context.
- Carry out a test by obtaining as much information as you can about the \(p\)-value.
- The client requests that you find which brands are significantly different from each other. Do you think you are able to fulfill the client’s request? If you can, provide an answer. If you cannot, discuss why you are unable to fulfill the client’s request and if there are other steps necessary to fulfill the request.
Exercise 07
Consider independent samples from two normal populations: \[X_{1},\ldots,X_{n}\sim N\left(\mu_{1},\sigma_{1}^{2}\right),\ Y_{1},\ldots,Y_{m}\sim N\left(\mu_{2},\sigma_{2}^{2}\right),\] where \(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2}\) are unknown. Your task is to find the distribution of\[\frac{1}{\sigma_{1}^{2}}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}+\frac{1}{\sigma_{2}^{2}}\sum_{i=1}^{m}\left(Y_{i}-\overline{Y}\right)^{2}\] under different settings and under finite \(n\). If the distribution is known, specify the distribution completely. If the distribution is not known, write UNKNOWN. What is the distribution:
when \(\mu_{1}=\mu_{2}=\mu\) and \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\)?
when \(\mu_{1}\neq\mu_{2}\) and \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\)?
when \(\sigma_{1}^{2}\neq\sigma_{2}^{2}\)?