AMS 310 Spring 2024 HW 1

AMS 310 Homework 1 Spring 2024 Prof. F. Rispoli Show all steps and do not just give the final answers. Material is based on Chapters 1 & 2 from Ahn. Please upload your homework to Brightspace. You have two attempts to upload correctly for each homework. Show work for each problem. Solutions to problems 1 and 2 must be typed (you can use Word or any other program for typing). The others can be neatly hand-written. But it is recommended that you type up solutions to all problems. There are 12 problems. Problems 1 and 2 are worth 10 points. The other 10 problems are worth 8 points. In problems 1 and 2, 5 points per problem will be deducted if the solution is not typed. 1. The monthly average New York City temperature in Fahrenheit are given below. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec High 35 36 45 61 71 79 88 83 75 64 54 43 Low 22 25 30 45 54 64 71 68 61 50 42 28 a) Find the mean the Low and the High temperatures. Round to one decimal place High: (35+36+45+61+71+79+88+83+75+64+54+43) / 12 = 61.2 Low: (22+25+30+45+54+64+71+68+61+50+42+28) / 12= 46.7 b) Draw a time series plot of the average Low and High temperatures. Make one chart with both graphs. Use technology (R or Excel). Jan Feb M ar Apr May Jun Jul Aug Sep Oct Nov Dec 0 10 20 30 40 50 60 70 80 90 100 Time Series Plot Of Low and High Temp High Low

c) Compute the sample standard deviation of the Low temperatures using the short cut formula. 484+625+900+2025+2916+4096+5041+4624+3721+2500+1764+784=29480 ΣX: 560; ΣX²:29480 s² = [29480-(560²/12)]/11= √304.24 = 17.4 d) Create a side-by-side box plot for the High temperatures vs. Low temperatures. Use technology (R or Excel).

3. Use the Iris data set in R a) Create a histogram for Petal Width. Copy and Paste the histogram into your document. Paste it as a Picture. b) Create a histogram for Sepal Width. Copy and Paste the histogram into your document. Paste it as a Picture.

4. The table below gives the types of errors discovered during a surgical set up. Error Type Frequency Wrong Supplier 45 Excess Count 67 Too Few Count 21 Wrong Size 25 Wrong Sterile Instrument Set 9 Missing Item 18 Damaged Item 6 Other 2 a) Construct a Pareto Chart for this data. Be sure to include a secondary axis used for cumulative percentages. Do this by using Excel or using R. b) Identify which error types account for roughly 80% of the errors. Excess Count, Wrong Supplier, Wrong Size, and Too Few Count

b) Create a side by side-by-side set of box plots for the three data sets. The box plots can be created either by Excel or R. 6. Consider the following data and answer the questions using R . Make sure your answer includes both the R command and the final answer/diagram. Copy and paste these into your Word document. 55 61 94 94 69 77 68 54 85 77 92 92 81 73 69 81 75 84 70 81 81 89 59 72 82 62 a) Create a histogram. > six=c(55,61,94,94,69,77,68,54,85,77,92,92,81,73,69,81,75,84,70,81,81,89,59,72,82,62) > hist(six) b) Find the sample mean and sample variance using R. > var(six) [1] 138.8385 > mean(six) [1] 76.03846 c) Find the 5 number summary using the quantile function in R. > summary(six) Min. 1st Qu. Median Mean 3rd Qu. Max. 54.00 69.00 77.00 76.04 83.50 94.00 d) Create a box plot, make sure to include a scale. > boxplot(six)

7. A boy has color blindness and has trouble distinguishing blue and green. There are 75 blue pens and 25 green pens mixed together in a box. Given that he picks up a blue pen, there is a 80% chance that he thinks it is a blue pen and a 20% chance that he thinks it is a green pen. Given that he picks a green pen, there is an 90% chance that he thinks it is a green pen and a 10% chance that he thinks it is a blue pen. Assume that the boy randomly selects one of the pens from the box. a) What is the probability that he picks up a blue pen and recognizes it as blue? .75 x .80 = 60% b) What is the probability that he chooses a pen and thinks it is blue? Picks up blue pen, thinks it's blue: .75 x .80 = 60% Picks up green pen, thinks it's green: .25 x .10 = 2.5% 60% + 2.5% = 62.5% b) Given that he thinks he chose a blue pen, what is the probability that he actually chose a blue pen? 0.60/0.625 = 0.96 = 96%

9. Answer the following questions: a) In the general education course requirement at a college, a student needs to choose one each from social sciences, humanities, natural sciences, and foreign languages. There are 5 social science courses, 6 humanity courses, 3 natural science courses, and 4 foreign language courses available for general education. How many different ways can a student choose general education courses from these four areas? 5 x 6 x 3 x 4 = 360 b) 4 people are chosen from a 25-member club for president, vice president, secretary, and treasurer. In how many different ways can this be done? 25 x 24 x 23 x 22 = 12650 c) A student is randomly choosing the answer to each of 5 multiple choice questions in a test. Each question has 4 possible answers. How many possible ways can the student answer the five questions? 4 5 = 1024 d) Using an ordinary deck of playing cards, how many possible ways can a hand of 8 cards be dealt? 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 = 3.0342338 x 10 13 e) Determine the probability that a bit string of length 10 contains exactly 4, 5 or 6 ones. P(x=4) + P(x=5) + P(x=6) [10! / 4! x 6!] x (1/2) 10 + 10! / 5! x 5!] x (1/2) 10 + [10! / 6! x4!] x (1/2) 10 = 0.65625

10. Suppose components A 1 , A 2 , B 1 , B 2 and C 1 operate independently in the system shown below. Assume that all the components in the system start operating at the same time and a component does not work again once it fails. The probability of functioning for each component is 0.90 except for C. The probability that C works is 0.98. The entire system works if A 1 or A 2 works, and B 1 or B 2 works, and C 1 works. Find the probability that the entire system works. Show your reasoning. Round the answer off to three decimal places. P (A U B) = P(A) + P(B) - P(A∩B) P (A 1 U A 2 ) = P(A 1 ) + P(A 2 ) - P(A 1 ∩A 2 ) 0.90 + 0.90 - 0.90 x 0.90 = 0.99 P (B 1 U B 2 ) = 0.99 P((A 1 U A 2 )∩(B 1 U B 2 )∩C 1 ) 0.99 x 0.99 x 0.98 = 0.960 11. Two 8-sided dice are tossed, and the sum is recorded. Find the probability that the sum is: a) Find the probability that the sum is 2. 8*8=64 possible outcomes The sum can range from 2 (if both dice show 1) to 16 (if both dice show 8). P(Sum is 2) = # of favorable outcomes/ Total # of Outcomes= 1/64 b) Find the probability that the sum is 8. Different ways to get sum of 8: (2, 6), (3, 5), (4, 4), (1, 7), (6, 2), (5, 3),(7,1) P(Sum is 8)= 7/64 c) Which sum has the largest probability of appearing? Possible sums: 2-16 16+2/2= 9 A 1 A 2 B 1 B 2 C 1