Homework 5

BSTA 550

Due: Thursday November 9, 2023 at 11pm
Published

November 9, 2023

Modified

November 6, 2023

Midterm Survey

With this homework, you need to complete the midterm survey. This will help me evaluate how well I am teaching.

There are 13 total questions. Only one question is required, so if you do not feel strongly or do not have suggestions, don’t feel like you need to answer. I want to hear your opinion! I want to make meaningful changes in the classroom that will help you! 

Don’t forget that this is 2% of your grade. This survey is also due November 9th at 11pm.

Directions

Please turn in this homework on Sakai. Please submit your homework in pdf format. You can type your work on your computer or submit a photo of your written work or any other method that can be turned into a pdf. Please let me know if you greatly prefer to submit a physical copy. We can work out another way for you to turn in homework.

Try to complete all of the problems listed below at some point this quarter! You may want to save some of them for studying later! Only turn in the ones listed in the “Turn In” column. Please submit problems in the order they are listed.

The more work you include that shows your thought process, the more I can give you feedback.

Chapter Turn In Extra Problems
12 NTB # 1-2 # 1, 2, 7, 11, 12, 15, 19, 25, 27, NTB # 4
13 (review) # 3, 4, 5, 6, 8, 9, 10, 17, 25
14 # 3, 7
15 NTB # 3 # 1, 5, 11, 18, 23, NTB # 5
16 TB # 7 # 3a-g, 8, 11, 21
17 TB # 9 # 3a-g, 6, 11, 12a-c, NTB # 6
18 TB # 20 # 1, 24, 26, 27
19 TB # 6 # 1, 18, 19
20 # 2, 3, 4

Non-textbook problems (NTB)

  1. Prove that for a r.v. \(X\) and constants \(a\) and \(b\), that \[\mathrm{Var}[aX+b]=a^2\mathrm{Var}[X].\]

  2. Let \(\bar{X}\) be the random variable for the sample mean, \(\bar{X}=\frac{\sum_{i=1}^nX_i}{n}\), where the \(X_i\) are i.i.d. random variables with common mean \(\mu\) and variance \(\sigma^2\).

    1. Find \(\mathbb{E}[\bar{X}]\).

    2. Find \(Var[\bar{X}]\).

  3. Let \(X_i\sim\) Binomial(\(n_i,p\)) be independent r.v.’s for \(i=1,\ldots,m\).

    1. What does the r.v. \(X=\sum_{i=1}^mX_i\) count, and what is the distribution of \(X\)? Make sure to specify the parameters of \(X\)’s distribution.

    2. Find \(\mathbb{E}[X]\). Make sure to show your work for (b) and (c). However, you may use without proof what you know about the mean and variance of each \(X_i\).

    3. Find \(Var[X]\).

      Extra Problems

  4. Let \(\hat{p}\) be the random variable for the sample proportion, \(\hat{p}=\frac{X}{n}\), where \(X\) is the number of successes in a random sample of size \(n\). Assume the probability of success is \(p\).

    1. Find \(\mathbb{E}[\hat{p}]\).

    2. Find \(Var[\hat{p}]\).

  5. Read the Washington Post article The amazing woman who can smell Parkinson’s disease - before symptoms appear (http://www.washingtonpost.com/news/morning-mix/wp/2015/10/23/scottish-woman-detects-a-musky-smell-that-could-radically-improve-how-parkinsons-disease-is-diagnosed/)

    Assuming Joy Milne does not have the ability to detect Parkinson’s disease via smell, answer the following questions:

    1. What is the probability of her correctly detecting Parkinson’s by smelling one t-shirt?

    2. What is the probability of her correctly detecting Parkinson’s in 12 out of 12 t-shirts?

  6. Let \(X_i\sim\) Negative Binomial(\(r_i,p\)) be independent r.v.’s for \(i=1,\ldots,m\).

    1. What does the r.v. \(X=\sum_{i=1}^mX_i\) count, and what is the distribution of \(X\)? Make sure to specify the parameters of \(X\)’s distribution.

    2. Find \(\mathbb{E}[X]\). Make sure to show your work for (b) and (c). However, you may use without proof what you know about the mean and variance of each \(X_i\).

    3. Find \(Var[X]\).

Some select answers

Selected answers (or hints) not provided at the end the book:

  • Chapter 12

    • # 2:  64.8

    • # 12:  1,096,357

  • Chapter 13

    • # 4:  (a) 260/9     (b) 2.833     (c) \(2.679\times 10^{-5}\)     (d) Same idea as (c) Replace 10’s with 100.     

    • # 6:  (a) \(p_X(x)=\binom{4}{x}.3^x .7^{4-x}\), for \(x=0,1,\ldots,4\)     (d) 0.3483     (e) 0.9163     (f) 0.0233     (g) 1

    • # 8:  (a) T     (b) F     (c) F     (d) F     (e) T     (f) T    (g) T

    • # 10:  (a) T     (b) T    (c) F     (d) T    (e) T     (f) F    (g) T     (h) T (nonnegative instead of positive)     (i) F

  • Chapter 15

    • # 18  (a) Bin(21,0.65)     (b) 4.78     

  • Chapter 16

    • # 8  (a) 14.28      (b) code below     (c) \(1.03\times 10^{-6}\)    (d) 10 questions: 91.43 minutes    
    1-pgeom(q = 18, prob = 0.07)
    [1] 0.2518698
    ## OR
pgeom(q = 18, prob = 0.07, lower.tail = F)
    [1] 0.2518698

  • Chapter 17

    • # 6   (a) 400, 87.18     (b) No     

    • # 12   (c) 0.8000

  • Chapter 18

    • # 24  (c) 0.8571

    • # 26  162,754.8

  • Chapter 19

    • # 6:  (c) 15.625     (d) 0.0486     (f) 0.0488

    • # 18:   100

  • Chapter 20

    • # 2:  (a) 0.0001     (b) Discrete since \(X\) has a finite number of possible values. Uniform since each outcome is equally likely.     (c) \(X\) = randomly selected 4-digit ID#; \(X=0000,0001,\ldots,9999\)     (d) 5000.5     (e) 8,333,333.25