Reflection 8

Exam 2 review

Computational questions (mandatory)

  1. Compute sample standard deviation by hand on the following data set:
x = 7, 12, 8, 6, 13, 12

  1. What is the area under a standard normal distribution (i.e. the percentage of data or probability) that lies to the left of z = -1.3?

  2. What is the area under a standard normal distribution (i.e. the percentage of data or probability) that lies to the right of z = -0.7?

  3. p = 0.1788 of the data in a normal distribution lies to the right of what z-score (1 pt.)?

  4. Iowa averages 34.8 tornadoes per year with a standard deviation of 22. Assuming a normal distribution and statistical independence, what is the probability that it receives between 22 and 32 tornadoes in a year?

Conceptual questions (optional)

  1. Explain the difference between mutually exclusive and independent events. You may use examples if you wish.

  2. Describe a use case for the standard deviational ellipse.

  3. Does standard distance (i.e., the multi-dimensional form of standard deviation) follow the 68-95-99.7 rule? Why or why not (2 pts.)?

  4. Describe the following distribution in terms of kurtosis, skewness, and similarity to a normal distribution. You may calculate these values if you wish to confirm your answer (the dataset is ggplot2::diamonds$depth).

library(ggplot2)

ggplot(diamonds) +
  geom_density(aes(depth), fill = "cyan")

  1. Describe the following distribution in terms of kurtosis, skewness, and similarity to a normal distribution. You may calculate these values if you wish to confirm your answer (the dataset is ggplot2::diamonds$carat).
library(ggplot2)

ggplot(diamonds) +
  geom_density(aes(carat), fill = "magenta")

  1. Suppose there is population of 10,000 observations which roughly follows a chi-square distribution with 4 degrees of freedom:
library(haffutils)

set.seed(2023)
x <- rchisq(1e4, df=4)

pretty_dens(x)

A sample of 1,000 values from this distribution will take on the shape of what distribution?

  1. Suppose there is population of 10,000 observations which roughly follows the additive inverse of chi-square distribution with 7 degrees of freedom:
library(haffutils)

set.seed(2023)
x <- 1 - rchisq(1e4, df=7)

pretty_dens(x)

Suppose you take 1,000 samples of 15 values each. The means of these samples will follow the shape of what distribution?

  1. What are the assumptions of using z-scores?

  2. Explain whether or not each of the following is an example of independent events:

  • Relative humidity at stations 500m apart from one another
  • Income at the census tract level
  • Heights of 100 randomly sampled individuals across the globe
  1. Explain why z-scores are inappropriate to use on the following dataset:
options <- c("Strongly disagree", "Disagree", "Neutral", "Agree", "Strongly agree")

set.seed(03262023)
results <- sample(options, size = 20, replace = TRUE)

print(results)
##  [1] "Disagree"          "Strongly disagree" "Agree"            
##  [4] "Strongly disagree" "Strongly agree"    "Agree"            
##  [7] "Disagree"          "Neutral"           "Strongly disagree"
## [10] "Strongly agree"    "Strongly agree"    "Strongly disagree"
## [13] "Neutral"           "Disagree"          "Strongly agree"   
## [16] "Strongly agree"    "Neutral"           "Neutral"          
## [19] "Neutral"           "Strongly agree"

Extra computational questions (optional)

  1. Compute population standard deviation by hand on the following data set:
x = 18, 12, 13, 14, 13, 11

  1. What is the area under a standard normal distribution (i.e. the percentage of data or probability) that lies to the right of z = -0.7?

  2. What is the area under a standard normal distribution (i.e. the percentage of data or probability) that lies to the left of z = -1.3?

  3. p = 0.87 of the data in a normal distribution lies to the left of what z-score (1 pt.)?

  4. Take a look at the treering dataset in R. What is the probability of observing a value between 1.2 and 1.8 (assume that this is a population)?