Note:
  • If it is a reverse question (given a percentage/proportion and asked to find a value of the variable) then you must include type=“q” in distrib().
  • If it is a “right-of” question then you must include lower.tail=FALSE in distrib().
  • All “between” questions require a three-step process with distrib() to answer. If it is a forward question then distrib() is used to find the areas to the left of the two given values and then those two areas are subtracted. If it is a reverse question then the area between must be converted to a left-of area and a right-of area which are then separately put into distrib() to compute two values (which are the final answer).
  • I strongly urge you to continue to identify whether the question is a “forward” or “reverse” question (which will identify whether you need to use type=“q” in distrib() or not) and a “left-of”, “right-of”, or “between” question (which will identify whether you need to use lower.tail=FALSE in distrib() or not and whether the answer is a three-step process to calculate).
  • Proportions are decimals; they are the value before you multiply by 100 to make a percentage.
  • Percentages should be rounded to one decimal. Values of “x” should be rounded to what makes sense for the problem (e.g., no decimals for discrete variables).

Carpenter Ants

  1. 4.8%.
  2. 1711 ants.
  3. 1268 ants.
  4. 94.2%.
  5. 0.01%.
  6. 1893 ants.
  7. Between 1016 and 1784 ants.

R Appendix.

distrib(1900,mean=1400,sd=300,lower.tail=FALSE)*100
distrib(0.15,mean=1400,sd=300,type="q",lower.tail=FALSE)
distrib(0.33,mean=1400,sd=300,type="q")
ab <- distrib(1900,mean=1400,sd=300)
a <- distrib(700,mean=1400,sd=300)
(ab-a)*100
distrib(300,mean=1400,sd=300)*100
distrib(0.05,mean=1400,sd=300,type="q",lower.tail=FALSE)
distrib(0.10,mean=1400,sd=300,type="q")
distrib(0.10,mean=1400,sd=300,type="q",lower.tail=FALSE)

SAT Scores

  1. SAT score is a continuous quantitative variable (this may be discrete, so I counted discrete as correct).
  2. The proportion of students not accepted by School A is 0.221.
  3. The percentage of students accepted by School B is 6.2%.
  4. The percentage of students accepted by School A but not School B is 71.7%.
  5. School C should set the acceptance criterion at 594.

R Appendix.

distrib(500,mean=550,sd=65)
distrib(650,mean=550,sd=65,lower.tail=FALSE)
ab <- distrib(650,mean=550,sd=65)
a <- distrib(500,mean=550,sd=65)
ab-a
distrib(0.25,mean=550,sd=65,type="q",lower.tail=FALSE)

Urban Deer Relocations

  1. The proportion of deer with home ranges between 0.2 and 0.4 km2 is 0.707.
  2. The proportion of deer with a home range greater than 0.32 km2 is 0.417.
  3. The home range such that 17% of the deer have a larger home range is 0.39 km2.
  4. The home range such that 32% of the deer have a smaller home range is 0.26 km2.
  5. The proportion of deer with a home range less than 0.4 km2 is 0.854.
  6. The most common 48% of home ranges are between 0.24 and 0.36 km2.

R Appendix.

ab <- distrib(0.4,mean=0.3,sd=0.095)
a <- distrib(0.2,mean=0.3,sd=0.095)
ab-a
distrib(0.32,mean=0.3,sd=0.095,lower.tail=FALSE)
distrib(0.17,mean=0.3,sd=0.095,type="q",lower.tail=FALSE)
distrib(0.32,mean=0.3,sd=0.095,type="q")
distrib(0.4,mean=0.3,sd=0.095)
distrib(0.26,mean=0.3,sd=0.095,type="q")
distrib(0.26,mean=0.3,sd=0.095,type="q",lower.tail=FALSE)







iPhone Battery Lifespan

  1. Number of charge-cycles is a discrete quantitative variable.
  2. The proportion of betteries that would be rated as “exceptional” is 0.006.
  3. The percentage of batteries that would be rated as better than “acceptably poor” but not “exceptional” is 88.8%.
  4. The new definition of “exceptional” (i.e., the top 10% of batteries) would be 426 charge-cycles.
  5. The new definition of “unacceptably poor” (i.e., the bottom 25% of batteries) would be 387 charge-cycles.

R Appendix.

distrib(450,mean=400,sd=20)
ab <- distrib(450,mean=400,sd=20)
a <- distrib(375,mean=400,sd=20)
ab-a
distrib(0.10,mean=400,sd=20,type="q",lower.tail=FALSE)
distrib(0.25,mean=400,sd=20,type="q")

Normal Distribution Characteristics II

  1. Below
    1. \(\mu\)=0, \(\sigma\)=1
    2. \(\mu\)=90, \(\sigma\)=8

Hand Calculations II

For Z~N(0,1)

  1. Z=1
  2. 2.5%
  3. 99.85%
  4. Between Z=-3 and 3.

Pollen Counts

Top questions

  1. A day in LaCrosse, WI n early September
  2. Pollen count (spores per cubic meter)
  3. Discrete quantitative
  4. \(\mu\)=40
  5. \(\sigma\)=8

Bottom questions

    1. reverse, right-of; (b) 50.3 spores per cubic meter.
    1. forward, left-of; (b) 0.894.
    1. forward, between; (b) 0.811.
    1. reverse, left-of; (b) 35.8 spores per cubic meter.
    1. reverse, between; (b) Between 31.7 and 48.3 spores per cubic meter.
    1. forward, right-of; (b) 0.734.

R Appendix.

distrib(0.1,mean=40,sd=8,type="q",lower.tail=FALSE)
distrib(50,mean=40,sd=8)
ab <- distrib(55,mean=40,sd=8)
a <- distrib(32,mean=40,sd=8)
ab-a
distrib(0.3,mean=40,sd=8,type="q")
distrib(0.15,mean=40,sd=8,type="q")
distrib(0.15,mean=40,sd=8,type="q",lower.tail=FALSE)
distrib(35,mean=40,sd=8,lower.tail=FALSE)
Note:
  • The IQR question below (#4) is the same as asking “what is the most common 50% of times to pass between the two points?”.
  • For question #6 below, the median is equal to the mean for a normal distribution (or any distribution that is perfectly symmetric).

Driving Speed

Top questions

  1. A car traveling between the two points
  2. Time to travel between the two points
  3. Continuous quantitative
  4. \(\mu\)=2.5
  5. \(\sigma\)=0.75

Bottom questions

    1. reverse, right-of; (b) 3.28 s
    1. forward, left-of; (b) 0.023.
    1. forward, between; (b) 0.905.
    1. reverse, between; (b) The IQR is from a Q1 of 1.99 s to a Q3 of 3.01 s.
    1. forward, right-of; (b) 0.000000000987.
    1. reverse, left-of or right-of; (b) 2.5 s.

R Appendix.

distrib(0.15,mean=2.5,sd=0.75,type="q",lower.tail=FALSE)
distrib(1,mean=2.5,sd=0.75)
ab <- distrib(4.5,mean=2.5,sd=0.75)
a <- distrib(1.5,mean=2.5,sd=0.75)
ab-a
distrib(0.25,mean=2.5,sd=0.75,type="q")
distrib(0.25,mean=2.5,sd=0.75,type="q",lower.tail=FALSE)
distrib(7,mean=2.5,sd=0.75,lower.tail=FALSE)

Turkey Spur Length

  1. The spur length such that 30% of the turkeys have a longer spur length is 23 mm.
  2. The proportion of turkeys with spur lengths between 15 and 25 mm is 0.811.
  3. The proportion of turkeys with spur lengths greater than 30 mm is 0.007.
  4. The spur length such that 10% of turkeys have a smaller spur length is 16 mm.
  5. The proportion of urkeys with a spur length less than 18 mm is 0.217.
  6. The most common 80% of spur lengths are between 16 and 26 mm.

R Appendix.

distrib(0.3,mean=20.9,sd=3.7,type="q",lower.tail=FALSE)
ab <- distrib(25,mean=20.9,sd=3.7)
a <- distrib(15,mean=20.9,sd=3.7)
ab-a
distrib(30,mean=20.9,sd=3.7,lower.tail=FALSE)
distrib(0.1,mean=20.9,sd=3.7,type="q")
distrib(18,mean=20.9,sd=3.7)
distrib(0.1,mean=20.9,sd=3.7,type="q")
distrib(0.1,mean=20.9,sd=3.7,type="q",lower.tail=FALSE)