Stat 3011 Midterm 1 (Class Part)

Problem 1

All of this is summarized in the box on p. 25 in Wild and Seber.

On the first issue (what is the difference),

• In an experiment, the experimenter determines which experimental units receive which treatments (ideally by randomized assignment to treatment groups).
• In an observational study, which subject gets which treatment is outside the experimenter's control. Perhaps the subjects or their doctors choose. Perhaps it just happens (exposure to a toxic substance in the environment, for example).

On the second issue (what is the implication)

• A properly designed and executed experiment can reliably demonstrate causation.
• An observational study can only suggest possible causes. It cannot reliably establish causation. A properly designed and executed experiment is needed to follow up results implied by an observational study.

Problem 2

(a)

The mean is
$$\begin{align*}\sum_x x \mathop{\rm pr}\nolimits(x) & = 1 \cdot \frac{1}{7} + ... ...1 + 2 + 3 + 4 + 5 + 6 + 7}{7} \\ & = \frac{28}{7} \\ & = 4 \end{align*}$$

(b)

The standard deviation is

$$\mathop{\rm sd}\nolimits(X) = \sqrt{E\{(X - \mu)^2\}}$$

where $\mu$ is the mean calculated in part (a), and
$$\begin{align*}E\{(X - \mu)^2\} & = \sum_x (x - \mu)^2 \mathop{\rm pr}\nolimits... ...9 + 4 + 1 + 0 + 1 + 4 + 9}{7} \\ & = \frac{28}{7} \\ & = 4 \end{align*}$$
Thus $\mathop{\rm sd}\nolimits(X) = \sqrt{4} = 2$.

Problem 3

(a)

By the multiplication rule (using the assumed statistical independence)

$$\mathop{\rm pr}\nolimits(\text{no accidents in 30 days}) = \mathop{\rm pr}\nolimits(\text{no accident in one day})^30$$

So in order to answer this question we have to first answer the subsidiary question: what is the latter probability? By the complement rule
$$\begin{align*}\mathop{\rm pr}\nolimits(\text{no accident in one day}) & = 1 - ... ...text{an accident in one day}) \\ & = 1 - 0.002 \\ & = 0.998 \end{align*}$$
Thus

$$\mathop{\rm pr}\nolimits(\text{no accidents in 30 days}) = 0.998^30 = 0.941708$$

(b)

The events in parts (a) and (b) of this problem are complementary, so by the complement rule each is one minus the other

$$\mathop{\rm pr}\nolimits(\text{at least one accident in 30 days}) = 1 - 0.941708 = 0.058292$$

Problem 4

Curve A: skewed, long right tail, unimodal.

Curve B: symmetric, unimodal.

Curve C: skewed, long right tail, biimodal.

Curve D: symmetric, biimodal.