Question

For the following situations. Please indicate a distribution that would be appropriate to model thesituation (choices restricted to those we talked about). Additionally, if there is any information about theparameter values, include those as well.

i) A pesticide is being tested on groups of plants. 10 experiments are completed. Within each experi-ment a plant is sprayed with the pesticide. After one week, the plant is assessed and categorized aseither dead or alive. Pesticides with similar chemicals have a 95% survival rate.
ii) An athletic trainer is investigating the average mile time for high school cross country runners. Helooks at the data and observes that it is approximately bell shaped with an average of 5 minutes and45 seconds.iii) An economist is looking at US income data. He would like to use the Gaussian distribution, butnotices the data is heavily skewed.iv) A mental health practitioner is examining the amount of time (in months) that patients stick aroundin rehabilitation facilities. They will treat the times as continuous. Literature says that the averagesurvival time is 0.75 months.v) An engineer is looking to model the measurement error of a new instrument. The error looks bellshaped and approximately symmetric around 0.vi) A popular game show, ”Deal or No Deal,” has 26 cases (numbered 1 to 26) that contain variousamounts from $1 to $1,000,000. The game starts by the contestant picking one of the cases to holdonto. We are interested in a contestants initial selection.

Answer 1

i)

Answer:

binomial distribution

Parameters: n = 10, p = 0.95

Explanation:

When we want to calculate the probability of the number of success in a fixed number of trails (Bernoulli's trials i.e. two possible outcomes, success/failure), the binomial distribution follows.

Hence the sample size of 10 experiments will have a binomial distribution with parameters n and p.

n = number of trials = 10

p = probability of success = 0.95

ii)

Answer:

Normal distribution

Parameters: mean, standard deviation.

Explanation:

The normal distribution is used to represent an approximately bell-shaped curve. The normal distribution is defined by two parameters, i) mean and ii) the standard deviation.

iii)

Answer:

Beta distribution

Parameters:

Explanation:

The beta distribution is used to represents the skewed Gaussian distribution with two shape parameters, alpha, and beta.

iv)

Answer:

Normal distribution

Parameters: mean, standard deviation.

Explanation:

According to the Central Limit Theorem, the sample distribution of the sample mean follows a normal distribution with parameters, i) mean and ii) the standard deviation.

In this scenario, we are interested in the average time a patient sticks around in rehabilitation facilities hence the mental health practitioner study a sample to test the hypothesis that whether the average survival time is 0.75 months.

v)

Answer:

Normal distribution

Parameters: mean = 0, standard deviation.

Explanation:

The normal distribution is used to represent an approximately bell-shaped curve. The normal distribution is defined by two parameters, i) mean and ii) the standard deviation.

vi)

Answer:

Beta distribution

Parameters:

Explanation:

In a "Deal or No Deal" game, if we draw a frequency distribution of winners with respect to the expected winning amount, the distribution of expected winning amount will be skewed to the right such that as the expected winning amount, increases, the frequency will decrease. Hence the beta distribution can be used to represent this skewed distribution.