Question
A dermatologist will conduct an experiment to investigate the effectiveness of a new drug to treat acne. The dermatologist has recruited 36 pairs of identical twins. Each person in the experiment has acne and each person in the experiment will receive either the new drug or a placebo. After each person in the experiment uses either the new drug or the placebo for 2 weeks, the dermatologist will evaluate the improvement in acne severity for each person on a scale from 0 (no improvement) to 100 (complete cure).
(a) Identify the treatments, experimental units, and response variable of the experiment.
- Treatments:
- Experimental units:
- Response variable:
Each twin in the experiment has a severity of acne similar to that of the other twin. However, the severity of acne differs from one twin pair to another.
(b) For the dermatologist’s experiment, describe a statistical advantage of using a matched-pairs design where twins are paired rather than using a completely randomized design.
(c) For the dermatologist’s experiment, describe how the treatments can be randomly assigned to people using a matched-pairs design in which twins are paired.
▶️Answer/Explanation
Ans:
- Treatments: the type of drive received (new or place bol)
- Experimental units: The people participating in this expriment
- Response variable: Level of improvement after two weeks
A matches pairs design is statistically advantageous in this experiment become it rescues the effect of initial ache severity os d Confounding Variable. Since each twin in a pair has a simile dene severity to theofler, it is more effective to determine the drug’s effectiveness when composing between twins, is opposed to a randomized design where different ache Severities mad hake it harder to determine how r. much of an effect the drug octudily had.
For each pair of identical twin flip a fair coin to determine which twin gets the experiments l treatment. Whichever twin does not get the experimental treatment, gets the placebo. instead.
Question
To increase morale among employees, a company began a program in which one employee is randomly selected each week to receive a gift card. Each of the company’s 200 employees is equally likely to be selected each week, and the same employee could be selected more than once. Each week’s selection is independent from every other week.
(a) Consider the probability that a particular employee receives at least one gift card in a 52-week year.
(i) Define the random variable of interest and state how the random variable is distributed.
(ii) Determine the probability that a particular employee receives at least one gift card in a 52 -week year. Show your work.
(b) Calculate and interpret the expected value for the number of gift cards a particular employee will receive in a 52-week year. Show your work.
(c) Suppose that Agatha, an employee at the company, never receives a gift card for an entire 52-week year. Based on her experience, does Agatha have a strong argument that the selection process was not truly random? Explain your answer.
▶️Answer/Explanation
Ans:
\(x=\) the number of \(g_i f t\) cards an employee recieves,
The randan variable is Binomial with \(p=0,005 \& n=52\).
\(B=\) Binary (Recieres or does nit recieve)
\(I=\) “Each week’s selection is indeparders fromeveryother we uk”
\(N=\) fixed number \(=52\)
\(S \rightarrow\) Set probability \(\rightarrow 0.005\)
(ii) \(\begin{aligned} p(x \geq 1) & =1-p(x \leq 0) \\ & =1-\left(\begin{array}{c}52 \\ 0\end{array}\right)(0.005)^{\circ}(1-.995)^{52} \\ & =1-0.7705 \\ & =0.2295 \text { probability of recieving at least } \\ & \text { one gift carding } 52 \text {-week year. }\end{aligned}\)
(b)
$
\begin{aligned}
& E(x)=n p=52(000) \\
& E(x)=0.26
\end{aligned}
$
If many, many random samples of 52 -w ert years we chooses, then there will approx. an average of \(.26 \mathrm{gift}\) cards a particular employee will recieve.
\(\begin{aligned} & \text { truly random? Explain your answer. } \\ & p(x=0)=\left(\begin{array}{c}52 \\ 0\end{array}\right)(0.005)^{\circ}(1-.995)^{62} \\ &=.7705\end{aligned}\)
There is approx, a 7705 chance of not getting a gif card cor on entire 52-week, this is a likely occurance to occur so Agatha does n’t have a strong arguement that the selection process is not truely random.