Week 3 Homework

3.2 Area under the curve, Part II. What percent of a standard normal distribution N(µ = 0,?? = 1) is found in each region? Be sure to draw a graph.

  1. Z > ???1.13
#A
x=seq(-3,3,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
x=seq(-3,1.13,length=200)
y=dnorm(x)
polygon(c(-3,x,1.13),c(0,y,0),col="gray")

pnorm(1.13)
## [1] 0.8707619
  1. Z < 0.18
#B
x=seq(-3,3,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
x=seq(-3,.18,length=200)
y=dnorm(x)
polygon(c(-3,x,.18),c(0,y,0),col="gray")

pnorm(.18)
## [1] 0.5714237
  1. Z > 8
#A
x=seq(-3,3,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
x=seq(-3,8,length=200)
y=dnorm(x)
polygon(c(-3,x,8),c(0,y,0),col="gray")

pnorm(8)
## [1] 1
  1. |Z| < 0.5
#A
x=seq(-3,3,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
x=seq(-3,.5,length=200)
y=dnorm(x)
polygon(c(-3,x,.5),c(0,y,0),col="gray")

pnorm(.5)
## [1] 0.6914625

3.4 Triathlon times, Part I. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30 - 34 group while Mary competed in the Women, Ages 25 29 group. Leo completed the race in 1:22:28 (4948 seconds), while Mary completed the race in 1:31:53 (5513 seconds). Obviously Leo ???nished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups: . The ???nishing times of the Men, Ages 30 - 34 group has a mean of 4313 seconds with a standard deviation of 583 seconds. . The ???nishing times of the Women, Ages 25 - 29 group has a mean of 5261 seconds with a standard deviation of 807 seconds. . The distributions of ???nishing times for both groups are approximately Normal. Remember: a better performance corresponds to a faster ???nish.

  1. Write down the short-hand for these two normal distributions.
#A
ls <- 4948
lmn <-4313
lsd<-583
ms <- 5513
mmn<-5261
msd<-807
(ls-lmn)/lsd
## [1] 1.089194
(ms-mmn)/msd
## [1] 0.3122677
  1. What are the Z-scores for Leo’s and Mary’s ???nishing times? What do these Z-scores tell you?
qnorm(pnorm(ls, lmn, lsd))
## [1] 1.089194
qnorm(pnorm(ms, mmn, msd))
## [1] 0.3122677
pnorm(ms, mmn, msd)
## [1] 0.6225814
  1. Did Leo or Mary rank better in their respective groups? Explain your reasoning.

Leo did better than Mary, his time was more than 1 standard deviation of his group, Mary was 1/3.

  1. What percent of the triathletes did Leo ???nish faster than in his group?

Leo did better 86% of the other in his group.

  1. What percent of the triathletes did Mary ???nish faster than in her group?

Mary did better than 62% of her group.

  1. If the distributions of ???nishing times are not nearly normal, would your answers to parts (b) - (e) change? Explain your reasoning.

The answers would change. THe pnorm stats apply to only normally distributed data.

3.18 Heights of female college students. Below are heights of 25 female college students.

  1. The mean height is 61.52 inches with a standard deviation of 4.58 inches. Use this information to determine if the heights approximately follow the 68-95-99.7% Rule.

Yes heights are within -3 to +3 standard deviations of the mean

mn <- 61.52
sdd<- 4.58
sd1<- mn+sdd
sd2<- mn+sdd*2
sd3<- mn+sdd*3
sd1m<-mn-sdd
sd2m<-mn-sdd*2
sd3m<-mn-sdd*3
sd1
## [1] 66.1
sd2
## [1] 70.68
sd3
## [1] 75.26
sd1m
## [1] 56.94
sd2m
## [1] 52.36
sd3m
## [1] 47.78
  1. Do these data appear to follow a normal distribution? Explain your reasoning using the graphs provided below.

Yes the histogram is nearly distributed and the theoretical quantiles fall mostly within the line.

3.22 Defective rate. A machine that produces a special type of transistor (a component of computers) has a 2% defective rate. The production is considered a random process where each transistor is independent of the others.

  1. What is the probability that the 10th transistor produced is the ???rst with a defect?
choose(10,1)*.02 ^ 9*.98
## [1] 5.0176e-15
  1. What is the probability that the machine produces no defective transistors in a batch of 100?
choose(100,0)*.98 ^ 100
## [1] 0.1326196
  1. On average, how many transistors would you expect to be produced before the ???rst with a defect? What is the standard deviation?

On average of transistors you would produce 50 transistors before the 1st defect with a standard deviation of 0.1443075

  1. Another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. On average how many transistors would you expect to be produced with this machine before the ???rst with a defect? What is the standard deviation?

On average of transistors you would produce 20 transistors before the 1st defect with a standard deviation of 0.2353756

  1. Based on your answers to parts (c) and (d), how does increasing the probability of an event a???ect the mean and standard deviation of the wait time until success?

Increasing probabilty decreases the amount of transistors before a defective one is produced.

3.38 Male children. While it is often assumed that the probabilities of having a boy or a girl are the same, the actual probability of having a boy is slightly higher at 0.51. Suppose a couple plans to have 3 kids.

  1. Use the binomial model to calculate the probability that two of them will be boys.

Binomial probability would be 38%

  1. Write out all possible orderings of 3 children, 2 of whom are boys. Use these scenarios to calculate the same probability from part (a) but using the addition rule for disjoint outcomes. Con???rm that your answers from parts (a) and (b) match.
round((.51*.51*.49+.51*.49*.51+.49*.51*.51)*100,0)
## [1] 38
round(choose(3,2)*((.51^2)*.49)*100,0)
## [1] 38
  1. If we wanted to calculate the probability that a couple who plans to have 8 kids will have 3 boys, brie???y describe why the approach from part (b) would be more tedious than the approach from part (a).
Part B would be more tedious because it will require 56 calculations added together.

3.42 Serving in volleyball. A not-so-skilled volleyball player has a 15% chance of making the serve, which involves hitting the ball so it passes over the net on a trajectory such that it will land in the opposing team’s court. Suppose that her serves are independent of each other.

  1. What is the probability that on the 10th try she will make her 3rd successful serve?

She will have a13% probability to be successfull on 3rd serve.

  1. Suppose she has made two successful serves in nine attempts. What is the probability that her 10th serve will be successful?

She will have a 34% probability to be successfull on 10th serve after 2 successful serves in 9 tries

  1. Even though parts (a) and (b) discuss the same scenario, the probabilities you calculated should be di???erent. Can you explain the reason for this discrepancy? ####After the 9 serve you are not looking at 15% probabity of tenth being successful and are not getting probabily of 1 out of 1 server, verses 1 out of 9.