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. \(Z < -1.35\)
# find the value for x 
m <- 0
sd <- 1
z <- -1.35
x <- z * sd + m
x
## [1] -1.35
# Finding the norm for x
pnorm(x, m, sd)
## [1] 0.08850799

Answer: 8.85% of a normal standard distribution is found in the region.

  1. \(Z > 1.48\)
# finding x when Z> 1.48
z <- 1.48
x2 <- z * sd + m
x2
## [1] 1.48
1-pnorm(x2, m, sd)
## [1] 0.06943662
normalPlot(m,sd, bounds = c(x2, 1e+06), tails = FALSE)

Answer: 6.94% of a normal standard distribution is represented in the region.

  1. \(-0.4 < Z < 1.5\)
z <- c(-0.4,1.5)
x3 <- z * sd + m
x3
## [1] -0.4  1.5
pnorm(x3,m,sd)
## [1] 0.3445783 0.9331928
normalPlot(mean = 0, sd = 1, bounds = c(-0.4, 1.5), tails = FALSE)

Answer: 58.9% of a normal standard distribution is represented in the region.

  1. \(|Z| > 2\) #### Z >2 and -Z >2 or Z < -2
z <- c(-2,2)
x4 <- z * sd + m
x4
## [1] -2  2
pnorm(x4,m,sd)
## [1] 0.02275013 0.97724987
normalPlot(mean = 0, sd = 1, bounds = c(-2,2), tails = FALSE)

Answer: 95.4% of a normal standard distribution is represented in the region.

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.

Men, Ages 30 - 34 N(μ = 4313, σ = 583)
Women, Ages 25 - 29 N(μ = 5261, σ = 807)

  1. What are the Z-scores for Leo’s and Mary’s finishing times? What do these Z-scores tell you?
# Finding the z score for leo
#(Leo = 4948, mean = 4313, Standard deviation = 583)
Leo_x <- 4948
Leo_m <- 4313
Leo_sd <- 583
Leo_z <- (Leo_x - Leo_m)/(Leo_sd)
Leo_z
## [1] 1.089194
# Finding the z score for Mary
# (Mary = 5513, mean = 5261, standard deviation = 807)
Mary_x <- 5513
Mary_m <- 5261
Mary_sd <- 807
Mary_z <- (Mary_x - Mary_m)/(Mary_sd)
Mary_z
## [1] 0.3122677

This z scores shows that Leo was 1.089 times standard deviation away from the mean while Mary was 0.312 times standard deviation away from the mean. Out of these two, Mary was more closer to her age category mean.

  1. Did Leo or Mary rank better in their respective groups? Explain your reasoning.

Based on their Z scores, Mary was more close to her respective age category mean. Therefor she rank better in her respective group.

  1. What percent of the triathletes did Leo finish faster than in his group?
pnorm(Leo_z, lower.tail = FALSE)
## [1] 0.1380342

Answer: 13.8%

  1. What percent of the triathletes did Mary finish faster than in her group?
pnorm(Mary_z, lower.tail = FALSE)
## [1] 0.3774186

Answer: 37.7%

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

Z score will not change but part where we find the pnorm will change as finishing times are not nearly normal.

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.

# Use the DATA606::qqnormsim function

qqnormsim(heights)

check weather 68% lie within one standard deviation

# height mean
heights_mean <- mean(heights)
# height standard deviation
heights_SD <- sd(heights)
1-2*pnorm(heights_mean+heights_SD, heights_mean, heights_SD, lower.tail = FALSE)
## [1] 0.6826895

68%

check weather 95% lie within two standard deviation

1-2*pnorm(heights_mean+2*heights_SD, heights_mean, heights_SD, lower.tail = FALSE)
## [1] 0.9544997

95%

check weather 99.7% lie within three standard deviation

1-2*pnorm(heights_mean+3*heights_SD, heights_mean, heights_SD, lower.tail = FALSE)
## [1] 0.9973002

99.7%

Yes it does follow the 68-95-99.7% Rule

  1. Do these data appear to follow a normal distribution? Explain your reasoning using the graphs provided below.
hist(heights, probability = TRUE, xlab = "Heights", ylim = c(0, 0.1))
x <- 30:90
y <- dnorm(x = x, mean = heights_mean, sd = heights_SD)
lines(x = x, y = y, col = "blue")
abline(v = heights_mean, col = "red")

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?
  2. What is the probability that the machine produces no defective transistors in a batch of 100?
  3. On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation?
  4. 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?
  5. 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?

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.
  2. 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.
  3. 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).

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?
  2. Suppose she has made two successful serves in nine attempts. What is the probability that her 10th serve will be successful?
  3. 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?

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