Winter 2011 Final

Exam on WebCT.

Under construction

1Question 1¶

Let $X_1, \ldots, X_n$ be a random sample from the density function

$$f(x) = \begin{cases} \displaystyle \frac{4}{\theta} x^3 e^{-x^4/\theta} & \text{if } x > 0, \\ 0 & \text{if } x \leq 0,\end{cases}$$

where $\theta > 0$.

(a) Find a sufficient statistic for $\theta$.
(b) Find the maximum likelihood estimator $\hat \theta$ of $\theta$.
(c) Show that your MLE of part (b)¹ is unbiased.
(d) Is your MLE an MVUE? Why?

1.1Solution¶

1.2Accuracy and discussion¶

2Question 2¶

(a) Given $\alpha$ with $0 < \alpha < 1$, let $F_{\alpha, n_1, n_2}$ be such that

$$Pr\{Y > F_{\alpha, n_1, n_2}\} = \alpha$$

where $Y$ has the $F$-distribution with $n_1, n_2$ degrees of freedom. Show that $\displaystyle F_{1-\alpha,n_1, n_2} = \frac{1}{F_{\alpha, n_1, n_2}}$.
(b) Given independent random samples of sizes $n_1$ and $n_2$ from the population $N(\mu_1, \sigma_1^2)$ and $N(\mu_2, \sigma_2^2)$ respectively, use the pivot

$$\frac{s_1^2 / \sigma_1^2}{s_2^2 / \sigma_2^2}$$

to construct a $(1-\alpha) \times 100\%$ confidence interval for $\sigma_2^2/\sigma_1^2$.
(c) Two soft drink bottling machines 1 and 2 are to be compared. Five ten ounce bottles are filled by each machine and the resulting contents in fluid ounces were 8.0, 8.3, 7.2, 7.7, 8.1 for machine 1 and 6.0, 9.3, 7.3, 6.7, 8.9 for machine 2. Assuming that the data come from normal populations $N(\mu_1, \sigma_1^2)$ and $N(\mu_2, \sigma_2^2)$, find a 90% confidence interval for $\sigma_2/\sigma_1$.

2.1Solution¶

Later

2.2Accuracy and discussion¶

Later

3Question 3¶

1200 U.S. stores are classified according to type and location, with the following results:

Observed cell frequencies

Some expected cell frequencies were calculated and are given in parentheses. Using level 0.05 of significance, test the hypothesis that type and location are independent.
(b) A group of rats, one by one, proceed down a ramp to one of five doors, with the following results:

Are the data sufficient to indicate that the rats show a preference for certain doors? That is, test the hypotheses

$$\begin{align*}H_0 & : p_1 = p_2 = p_3 = p_4 = p_5 = \frac{1}{5} \\ H_1 & : \text{not } H_0,\end{align*}$$

where $p_i$ = probability of choosing door $i$. Use $\alpha = 0.01$.

3.1Solution¶

3.2Accuracy and discussion¶

4Question 4¶

An experiment was conducted to determine the effect of three methods A, B and C of soil preparation on the first year growth of pine seedlings. Four locations 1, 2, 3, and 4 were selected, and each location was divided into three plots. A randomised block design was employed using locations as blocks. On each plot, the same number of seedlings was planted, and the average first-year growth in centimetres was recorded. These observations are given in the following table.

(a) Complete the following ANOVA table.

(b) Do the data provide sufficient evidence to indicate differences in the mean growth for the three soil preparations? Use $\alpha = 0.05$.
(c) Is there evidence to indicate differences in the mean growth for the four locations? Use $\alpha = 0.05$.

4.1Solution¶

4.2Accuracy and discussion¶

5Question 5¶

We observe a binomial random variable $X$ with parameters $n$ and $\theta$, i.e.

$$P[X = k] = \left ( \frac{n}{k} \right ) \theta^k(1-\theta)^{n-k}, \quad k = 0, 1, \ldots, n.$$

(a) Use the Neyman-Pearson lemma to find a most powerful critical region of size $\alpha$ for testing

$$\begin{align*} H_o\,:\, & \theta = \theta_0 \\ H_1 \,:\, & \theta = \theta_1\end{align*}$$

where $\theta_1 > \theta_0$.
(b) Suppose that $n = 20$ an we want to test

$$\begin{align*} H_o\,:\, & \theta = 0.3 \\ H_1 \,:\, & \theta = 0.5\end{align*}$$

at level $\alpha = 0.05$. What is the critical region?

5.1Solution¶

5.2Accuracy and discussion¶

6Question 6¶

The median sale prices for new single-family houses is given in the following table for the eight years 1972 through 1979 (i.e. year 3=1974).

The model $y = \beta_0 + \beta_1 x + \epsilon$ is to be fitted to this data.

(a) Find least squares estimates of $\beta_0$ and $\beta_1$. Hint: $\bar y = 43.3625$, $S_{xy} = 203.35$, $S_{yy} = 1460.25$.
(b) Is there sufficient evidence to indicate that the median sales price increased over the period from 1972 through 1979? Use $\alpha = 0.01$. Recall that

$$t = \frac{(\hat \beta_1 - \beta_1) \sqrt{\displaystyle S_{xx}}}{\hat \sigma}, \quad \hat \sigma^2 = \frac{SSE}{n-2}, \quad SSE = S_{yy} - \hat \beta_1 S_{xy}.$$

(c) Find a 99% confidence interval for $\beta_1$.

6.1Solution¶

Later

6.2Accuracy and discussion¶

Later