Question 3 to 5

(ns assignments.hw1.q3-5
  (:require
   [assignments.hw1.utils :refer :all]
   [fastmath.core :as m]
   [fastmath.random :as rand]))

---

Question 3

---

3) Assume that healthy human body temperatures are normally distributed with a mean of 98.02 degrees Fahrenheit and a standard deviation of 0.62 degrees Fahrenheit. A hospital uses 101 degrees Fahrenheit as the lowest body temperature (laser body temperature reading) criterion of having a fever. What percentage of healthy persons would be misclassified as having a fever with this criterion? Explain.

(let [threshold 101 mean-temp 98.02 sd-temp 0.62]
  (norm-plot-threshold threshold mean-temp sd-temp :right))

(let [mu 98.02 sd 0.62 x 101
      z (x->z x mu sd)
      prob (pnorm z)
      p (- 1 prob)]
  (answer
   (str "Percentage of healthy persons misclassified as having a fever: "
        (* 100 (m/approx p)) "%")))

Percentage of healthy persons misclassified as having a fever: 0.0%

Question 4

---

4) In Question 3, if a physician wants to select a minimum temperature with a rate of misclassifying healthy patients at 10%. What should that criterion be?

The inv-normal function calculates the inverse cumulative distribution function for a standard normal distribution:

\[\Phi^{-1}(p) = \mu + \sigma z\]

This is used to find the value corresponding to a given probability in a normal distribution.

The z->x function converts a value from a standard normal distribution to a normal distribution:

\[x = \mu + \sigma z\]

This is used to find the value corresponding to a given probability in a normal distribution.

(comment
  (defn norm-plot-pval
    "Creates a ggplot2 plot for the temperature distribution with p-value shading."
    [p-val mean-temp sd-temp direction]
    (let [z-score (inv-normal (if (= direction :right) (- 1 p-val) p-val))
          threshold (z->x z-score mean-temp sd-temp)
          xlim [(- mean-temp (* 5 sd-temp)) (+ mean-temp (* 5 sd-temp))]]
      (-> (ggplot :data (tc/dataset {:x xlim}) (aes :x 'x))
          (r+ (stat_function :fun dnorm :args [mean-temp sd-temp]
                             :geom "area" :fill "lightblue" :alpha 0.7)
              (stat_function :fun dnorm :args [mean-temp sd-temp]
                             :xlim (if (= direction :right)
                                     [threshold (second xlim)]
                                     [(first xlim) threshold])
                             :geom "area" :fill "red" :alpha 0.3)
              (geom_vline :xintercept threshold :color "red" :linetype "dashed")
              (geom_text :x threshold :y 0 :angle 90 :hjust -0.5 :vjust -0.5
                         :label (str "Threshold: " (format "%.2f" threshold)))
              (labs :title "Distribution of X"
                    :subtitle (str "Percent of data in red: " (* 100 p-val) "%")
                    :x "X"
                    :y "Density")
              (scale_x_continuous :limits xlim)
              (theme_minimal))
          plot->svg))))

(let [mu 98.02 sd 0.62 p-val 0.1]
  (norm-plot-pval p-val mu sd :right))

(defn rnorm [n mu sd]
  (let [dist (rand/distribution :normal {:mu mu :sd sd})]
    (repeatedly n #(rand/sample dist))))

(let [mu 98.02 sd 0.62 p 0.1 one-minus-p (- 1 p)
      z (inv-normal one-minus-p)
      x (z->x z mu sd)
      samples (rnorm 1000 mu sd)]
  (answer
   (str "Minimum temperature with a rate of misclassifying healthy patients at 10%: "
        (m/approx x))))

Minimum temperature with a rate of misclassifying healthy patients at 10%: 98.81

Question 5

---

5) A financial analyst states that the price of a long-term $1000 government bond (one year after purchasing) is normally distributed with an expected value $980 and a standard deviation $40.

a) Find the chance that the price is more than $1000 one year after purchasing.

(let [mu 980 sd 40 x 1000]
  (norm-plot-threshold x mu sd :right))

(let [mu 980 sd 40 x 1000
      z (x->z x mu sd)
      prob (pnorm z)
      p (- 1 prob)]
  (answer
   (str "Probability that the price is more than $1000 one year after purchasing: "
        (* 100 (m/approx p 4)) "%")))

Probability that the price is more than $1000 one year after purchasing: 30.85%

b) What is the chance that the price is between $960 and $1060 after purchase for one year?

(let [mu 980 sd 40 x1 960 x2 1060]
  (norm-plot-thresh-btw x1 x2 mu sd))

(let [mu 980 sd 40 x1 960 x2 1060 z1 (x->z x1 mu sd)
      z2 (x->z x2 mu sd) p1 (pnorm z1) p2 (pnorm z2) p (- p2 p1)]
  (answer
   (str "Probability that the price is between $960 and $1060 after purchase for one year: "
        (* 100 (m/approx p 3)) "%")))

Probability that the price is between $960 and $1060 after purchase for one year: 66.9%

source: src/assignments/hw1/q3_5.clj