Area under the curve, Part I. (4.1, p. 142) What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Be sure to draw a graph.

  1. \(Z < -1.35\)
  2. \(Z > 1.48\)
  3. \(-0.4 < Z < 1.5\)
  4. \(|Z| > 2\)

## [1] 0.08850799
## [1] 0.06943662
## [1] 0.5886145
## [1] 0.04550026

Triathlon times, Part I (4.4, p. 142) 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 finished 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:

Remember: a better performance corresponds to a faster finish.

  1. Write down the short-hand for these two normal distributions. Mens: X ~N(4313, 583)

Womens: X ~N(5261, 807)

  1. What are the Z-scores for Leo’s and Mary’s finishing times? What do these Z-scores tell you?

The Z score tells you the relationship of the times to the mean. Leo’s time is almost exactly 1 sd from the mean.

#Z = (x-m)/sd

(Leo = (4948 -  4313)/583)
## [1] 1.089194
(Mary = (4948 -  5261)/807) 
## [1] -0.3878563
  1. Did Leo or Mary rank better in their respective groups? Explain your reasoning.

Leo did better because his Z score is larger in magnitude and positive in direction.

  1. What percent of the triathletes did Leo finish faster than in his group?
pnorm(1.089194, mean = 0, sd = 1)
## [1] 0.8619658
  1. What percent of the triathletes did Mary finish faster than in her group?
pnorm(0.3122677, mean = 0, sd = 1)
## [1] 0.6225815
  1. If the distributions of finishing times are not nearly normal, would your answers to parts (b) - (e) change? Explain your reasoning.

I think that the Z score is still useful as long as the sd is still close to the sd if the distribution was normal. If the sd is wildly different than the Z score is going to be inaccurate.

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

\[ \stackrel{1}{54}, \stackrel{2}{55}, \stackrel{3}{56}, \stackrel{4}{56}, \stackrel{5}{57}, \stackrel{6}{58}, \stackrel{7}{58}, \stackrel{8}{59}, \stackrel{9}{60}, \stackrel{10}{60}, \stackrel{11}{60}, \stackrel{12}{61}, \stackrel{13}{61}, \stackrel{14}{62}, \stackrel{15}{62}, \stackrel{16}{63}, \stackrel{17}{63}, \stackrel{18}{63}, \stackrel{19}{64}, \stackrel{20}{65}, \stackrel{21}{65}, \stackrel{22}{67}, \stackrel{23}{67}, \stackrel{24}{69}, \stackrel{25}{73} \]

  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.

This data is very close to 68-95-99.7

heights <- c(54, 55, 56, 56, 57, 58, 58, 59, 60, 60, 60, 61, 
            61, 62, 62, 63, 63, 63, 64, 65, 65, 67, 67, 69, 73)
heights <- data.frame(heights)
heights$sd1 <- as.numeric(heights$heights > (61.52 - 4.58) & heights$heights < (61.52 + 4.58))
heights$sd2 <- as.numeric(heights$heights > (61.52 - 2*4.58) & heights$heights < (61.52 + 2*4.58))
heights$sd3 <- as.numeric(heights$heights > (61.52 - 3*4.58) & heights$heights < (61.52 + 3*4.58))

heights %>% summarise(mean(sd1), mean(sd2), mean(sd3))
  1. Do these data appear to follow a normal distribution? Explain your reasoning using the graphs provided below.

This graph looks normally distributed and the Q-Q plot is almost a horizonal line.

Based on the Q-Q simulations, it still looks normally distributed.

# Use the DATA606::qqnormsim function
heights <- data.frame(heights)
openintro::qqnormsim(data = heights, sample = heights)

Defective rate. (4.14, p. 148) 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 first with a defect?
#Probability of 9 non-events * prob of event
(((0.98)^9)*(0.02))
## [1] 0.01667496
  1. What is the probability that the machine produces no defective transistors in a batch of 100?
#Probability of 100 non-events
((0.98)^100)
## [1] 0.1326196
  1. On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation? Mean
#mean
(1/0.02)
## [1] 50
  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 first with a defect? What is the standard deviation?
#mean
(1/0.05)
## [1] 20
  1. Based on your answers to parts (c) and (d), how does increasing the probability of an event affect the mean and standard deviation of the wait time until success?

The mean and sd goes down as the probability increases

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.
0.51*0.51*0.49
## [1] 0.127449
  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. Confirm that your answers from parts (a) and (b) match.
0.49*0.51*0.51
## [1] 0.127449
0.51*0.51*0.49
## [1] 0.127449
0.51*0.49*0.51
## [1] 0.127449
  1. If we wanted to calculate the probability that a couple who plans to have 8 kids will have 3 boys, briefly describe why the approach from part (b) would be more tedious than the approach from part (a).

The approach in part B would be more tedious because you have to calculate all possible combinations of 8 choose 3

Serving in volleyball. (4.30, p. 162) 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?
((0.85)^7*(0.15)^2)*(0.15)
## [1] 0.001081948
  1. Suppose she has made two successful serves in nine attempts. What is the probability that her 10th serve will be successful?
(0.15)
## [1] 0.15
  1. Even though parts (a) and (b) discuss the same scenario, the probabilities you calculated should be different. Can you explain the reason for this discrepancy?

part a cares about all 10 events.

part b only cares about the 10th event.