Question

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Problem 1 For the following situations. Please indicate a distribution that would be appropriate to model the situation (choices restricted to those we talked about). Additionally, if there is any information about the parameter 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 as either 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. He looks at the data and observes that it is approximately bell shaped with an average of 5 minutes and 45 seconds. iii) An economist is looking at US income data. He would like to use the Gaussian distribution, but notices the data is heavily skewed. iv) A mental health practitioner is examining the amount of time (in months) that patients stick around in rehabilitation facilities. They will treat the times as continuous. Literature says that the average survival time is 0.75 months. v) An engineer is looking to model the measurement error of a new instrument. The error looks bell shaped and approximately symmetric around 0. vi) A popular game show, "Deal or No Deal," has 26 cases (numbered 1 to 26) that contain various amounts from $1 to $1,000,000. The game starts by the contestant picking one of the cases to hold onto. We are interested in a contestants initial selection. +

Answer 1

(i)

A pesticide is being tested on a group of 10 plants.

Let X denote the number of plants alive.

It is given that the pesticide has 95% survival rate.

Hence, X follows Binomial distribution with n = 10 and p = 0.95

(ii)

Let Y denote the average mile time for high school cross country runners. It is observed that the distribution is bell shaped with average 5 minutes and 45 seconds, that is 5.75 minutes.

We know that bell shaped distribution means Normal distribution.

Hence, Y follows Normal distribution with mean = 5.75 minutes.

(iii)

It is given that US income data may follow Gaussian distribution( that is Normal distribution), but it is heavily skewed.

Hene, US income data can be said to follow Skewed Normal distribution with non-zero shape parameter.

#for our usual normal distribution the shape parameter is zero.

(iv)

Let Z denote the amount of time that patients stick around in rehabilitation facilities.

Z is continuous random variable. According to literature the average survival time is 0.75 months.

We know that exponential distribution is used to model survival time.

Hence, Z follows exponential distribution with parameter $\lambda$ = 1/mean = 1/0.75 = 1.3333

(v)

Let W denote the measurement error of an instrument. It follows bell shaped distribution that is normal distribution symmetric around 0.

Hence, W follows Normal distribution with mean = 0.

(vi)

Let X denote the contestant's initial selection.

The game show has 26 cases(numbered from 1 to 26). Contenstant has 26 choices with each choice has equal chance of selection.

Hence, X follows (1,26) discrete uniform distribution with parameter n = 26-1+1 = 26.

I hope you find the solution helpful. If you have any doubt then feel free to ask in the comment section.

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