HWK4

Stat 371 Homework #4 Student’s Name Here • Submit your homework to Canvas by the due date and time. Email your lecturer if you have extenuating circumstances and need to request an extension. • If an exercise asks you to use R, include a copy of all relevant code and output in your submitted homework file. You can copy/paste your code, take screenshots, or compile your work in an Rmarkdown document. • If a problem does not specify how to compute the answer, you many use any appropriate method. I may ask you to use R or use manual calculations on your exams, so practice accordingly. • You must include an explanation and/or intermediate calculations for an exercise to be complete. • Be sure to submit the HWK4 Autograde Quiz which will give you ~20 of your 40 accuracy points. • 50 points total: 40 points accuracy, and 10 points completion Discrete Random Variables Exercise 1. A chemical supply company ships a certain solvent in 10-gallon drums. Let X represent the number of drums ordered by a randomly chosen customer. Assume X has the following probability mass function (pmf). The mean and variance of X are : µ X = 2 . 3 and σ 2 X = 1 . 81 : X P(X=x) 1 0.4 2 0.2 3 0.2 4 0.1 5 0.1 a. Calculate P ( X ≤ 2) and describe what it means in the context of the problem. b. Let Y be the number of gallons ordered, so Y = 10 X . Find the probability mass function of Y. Y P(Y=y) ? ? ? ? ? ? ? ? ? ? c. Calculate the mean number of gallons ordered µ Y . d. Calculate the standard deviation of the number of gallons ordered, σ Y . 1

Normal Random Variables Exercise 2. Weights of female cats of a certain breed (A) are well approximated by a normal distribution with mean 4.1 kg and standard deviation of 0.6 kg W A ∼ N (4 . 1 , 0 . 6 2 ) . a. What proportion of female cats of that breed (A) have weights between 3.7 and 4.4 kg? b. A female cat of that breed (A) has a weight that is 0.5 standard deviations above the mean. What proportion of female cats of that breed (A) are heavier than this one? c. How heavy is a female cat of this breed whose weght is on the 80th percentile? d. What is the IQR of weights for female cats of this breed using the normal distribution approximation? e. Females from another breed of cats (breed B) have weights well approximated by a normal distribution with mean 10.6 lb and standard deviation of 0.9 lb W B.lb ∼ N (10 . 6 , 0 . 9 2 ) . Transform the weights of cat breed B into kilograms using the conversion: 1 lb ≈ 0.454 kgs. You can use the transformation: W B = 0 . 454( W B.lb ) . Compare the shape, center, and spread of the two breeds. Sampling Distributions Exercise 3. A serving of breakfast cereal has a sugar content that is well approximated by a Normal random variable X with mean 13 g and variance 1 . 3 2 g 2 . We can consider each serving as an independent and identical draw from X. a. In what percent of servings will the sugar content be above 13.3 g? b. What is the probability that a randomly chosen serving will have a sugar content between 13.877 and 12.123? What do we call the difference: 13.877-12.123=1.754? c. Calculate the probability that in 6 servings, only 1 has a sugar content below 13 g. d. Describe the sampling distribution for the mean sugar content of 6 servings ¯ X . e. What is the interquartile range of the sampling distribution for the sample mean ¯ X when n=6? Is that value larger or smaller than the IQR implied in part (b)? Why do the relative sizes of the IQRs make sense? f. What is the probability that the mean sugar content in 6 servings is more than 13.3 g ? g. Is it more or less likely that the mean sugar content is above 13.3 g in 10 servings or 6 servings (as computed in f)? Can you explain it without actually computing the new probability? h. Suppose each cereal box of this type contains 10 servings and consider the total sugar content in each box as a sum of 10 iid random draws from X ∼ N (13 , 1 . 3 2 ) . If you were to eat a whole box of cereal, above what total sugar content would you consume with 95% probability? Show and briefly explain your calculations. Exercise 4. You will be comparing the sampling distributions for two different estimators of σ , the population standard deviation. When trying to estimate the standard deviation of a population ( σ ) from a sample we could use: s 1 = sssssss QQQQQQQ ( X − ¯ X ) 2 n − 1 or s 2 = rrrrrrr QQQQQQQ ( X − ¯ X ) 2 n 2