Download Math 1780.001 Homework: Probability and Statistics and more Assignments Mathematics in PDF only on Docsity!
Math 1780.001 (Cherry) Homework for Wednesday, July 22
Reading. Work carefully through Examples 3.5 and 3.6 on pages 82–83. Read sections 3.3 and 3.4 and work carefully through the examples with the book. Things will start to seem hard now.
Practice Problems. Work out the solutions to the following problems in a notebook and check your answers in the back of your text book. YOU WILL NOT TURN THESE PROBLEMS IN for a grade. They are only to help you study. Note however that these problems may appear verbatim on the weekly tests.
3.20–3.22, 3.23, 3.24, 3.25, 3.26, 3.
Written assignment on the back
Homework to turn in Wednesday, July 22
Applications of Techbysheff’s Theorem
- An instructor gives an exam out of a possible 100 points. Suppose the mean exam score is 70 with a standard deviation of 5. (a) No matter what the distribution of scores, subject to the given mean and standard deviation, at least what percentage of the class must have scored in the range 55-85? (b) If the instructor deems that a score of 85 or above on this exam is an A grade, what is the largest percentage of the class that could have received an A, given this mean and standard deviation? (c) If the instructor wants at most 15% of the class to receive an A, what should he or she set the A cut-off score to be?
- Suppose a gambler plays a game of chance. Let X be the random variable associated to the gambler’s winnings (measured in dollars) depending on the outcome of the game. Suppose E(X) = − 0. 5 and V (X) = 0. 0625. What is the largest possible value for P (X ≥ 0), meaning what is the largest possible probability that the gambler will make a profit?
Binomial Distributions
- Let X denote a random variable that has a binomial distribution with p = 0. 3 and n = 3. Compute the following BY HAND using only a simple calculator and show work! (10 points) (a) P (X = 2) (b) P (X ≥ 2) (c) P (X ≤ 2) (d) E(X) (e) V (X)
- Let X denote a random variable that has a binomial distribution with p = 0. 6 and n = 15. Use Excel, Table 2 of your book, or a TI-83 or similar calculator to evaluate the following probabilities: (10 points) (a) P (X ≤ 6) (b) P (X ≥ 12) (c) P (X = 8) (d) P (4 ≤ X ≤ 10).
- In testing the lethal concentration of a chemical found in polluted water, researchers have determined that the chemical will kill 15% of the fish that were exposed to it for 24 hours. Suppose 50 fish are put in the tank with the chemical. (10 points) (a) What is the probability that more than half the fish survived? (b) What is the probability that exactly 45 fish survived? (c) What is the probability that 20–30 fish survived? (d) What is the expected number of fish to survive? (e) What is the standard deviation? Hint: For (d) and (e) use the formulas for expected value and variance on page 95.
Excel notes: Instead of using Table 2 in your book, you can use Excel. To get the numbers in Table 2 you type =BINOMDIST(k,n,p,TRUE). For example, =BINOMDIST(1,0.10,5,TRUE) gives the value 0. 919 from Table 2. This number is P (X ≤ 1). If you want P (X = 1), you could change the “TRUE” to “FALSE” to =BINOMDIST(1,0.10,5,FALSE) and you get 0. 329.
TI-83 notes: Instead of using Table 2 in your book, you can use a TI-83 calculator. To get the numbers in Table 2 you use the “Distribution menu” accessible using the “DISTR” key (located above “VARS”): 2nd VARS. Now, arrow down until you get to “binomcdf(” which stands for “binomial cumulative distribution function.” To get the values in Table 2 of your book, you enter: binomcdf(n,p,k). For example, binomcdf(5,0.10,1) gives the value 0. 919 from Table 2. This number is P (X ≤ 1). If you want P (X = 1), you could enter binompdf(5,0.10,1) and you get 0. 329. The name binompdf( stands for “binomial probability distribu- tion function.”
