STATS 67-The weight of adult male tigers follows a normal distribution

Subject: Mathematics    / Statistics
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Question

1. The weight of adult male tigers follows a normal distribution with mean µ =400 pounds and standard deviation ?=30 pounds.

a. Write out in integral form the probability that an adult male tiger will weigh between 370 and430 pounds. Use the pnorm(x, µ, ?) function (take difference of two of them) to compute this probability.

b. How much would a tiger need to weigh to have the chance or probability of weighing less than this tiger be 0.75 (tiger is heavier than 75% of all tigers)? Write out the integral and what it would be equal to and use qnorm() function in R to find this weight.

c. Now we have 2 tigers that are going to be randomly sampled. Let X1 be the weight of the firsttiger and X2 be the weight of the second tiger. What distribution does the sum of the weights,Y = X1 + X2, of these two tigers follow? What is the probability the weight of these two tigers isbelow 800 pounds (write out the form).

2. You are to model the daytime temperature of the Mojave desert during the summer time.Temperatures in the desert are modeled to follow a normal distribution with µ = 102 Fahrenheitiiand variance ?2 = 25 Fahrenheit.

a. Write out the integral form of the probability that the temperature will be between 97 to 107degrees Fahrenheit. Use pnorm function in R, pnorm(x, µ, ?) or use empirical rule to computenumeric value.

b. Say X is the random variable that denotes the temperature in Fahrenheit during the desertday. What distribution does Y=0.55X-17.6 follow (the temperature in Celsius)? State the type ofdistribution, and all parameters.

c. Now say you relax the normality assumption from the model, and you observe a random sampleof 25 days (X1, X2, …, X25 or to say Xi for i=1,2,3,…,25). What is the approximate distributionof the sample mean of these daily temperatures, X¯ =1nPni=1Xi? State the approximate type ofdistribution and all parameters.

d. You now transform each daily temperature from Fahrenheit to Celsius. Yi = 0.55Xi?17.6. Whatapproximate distribution does the sample mean in Celsius follow? That is to say the approximatedistribution of Y¯ =1nPni=1Yi. State the type and parameters.