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”
indistrib()
. -
If it is a “right-of” question then you must include
lower.tail=FALSE
indistrib()
. -
All “between” questions require a three-step process with
distrib()
to answer. If it is a forward question thendistrib()
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 intodistrib()
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”
indistrib()
or not) and a “left-of”, “right-of”, or “between” question (which will identify whether you need to uselower.tail=FALSE
indistrib()
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
- 4.8%.
- 1711 ants.
- 1268 ants.
- 94.2%.
- 0.01%.
- 1893 ants.
- 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
- SAT score is a continuous quantitative variable (this may be discrete, so I counted discrete as correct).
- The proportion of students not accepted by School A is 0.221.
- The percentage of students accepted by School B is 6.2%.
- The percentage of students accepted by School A but not School B is 71.7%.
- 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
- The proportion of deer with home ranges between 0.2 and 0.4 km2 is 0.707.
- The proportion of deer with a home range greater than 0.32 km2 is 0.417.
- The home range such that 17% of the deer have a larger home range is 0.39 km2.
- The home range such that 32% of the deer have a smaller home range is 0.26 km2.
- The proportion of deer with a home range less than 0.4 km2 is 0.854.
- 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
- Number of charge-cycles is a discrete quantitative variable.
- The proportion of betteries that would be rated as “exceptional” is 0.006.
- The percentage of batteries that would be rated as better than “acceptably poor” but not “exceptional” is 88.8%.
- The new definition of “exceptional” (i.e., the top 10% of batteries) would be 426 charge-cycles.
- 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
- Below
- \(\mu\)=0, \(\sigma\)=1
- \(\mu\)=90, \(\sigma\)=8
Hand Calculations II
For Z~N(0,1)
- Z=1
- 2.5%
- 99.85%
- Between Z=-3 and 3.
Pollen Counts
Top questions
- A day in LaCrosse, WI n early September
- Pollen count (spores per cubic meter)
- Discrete quantitative
- \(\mu\)=40
- \(\sigma\)=8
Bottom questions
- reverse, right-of; (b) 50.3 spores per cubic meter.
- forward, left-of; (b) 0.894.
- forward, between; (b) 0.811.
- reverse, left-of; (b) 35.8 spores per cubic meter.
- reverse, between; (b) Between 31.7 and 48.3 spores per cubic meter.
- 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
- A car traveling between the two points
- Time to travel between the two points
- Continuous quantitative
- \(\mu\)=2.5
- \(\sigma\)=0.75
Bottom questions
- reverse, right-of; (b) 3.28 s
- forward, left-of; (b) 0.023.
- forward, between; (b) 0.905.
- reverse, between; (b) The IQR is from a Q1 of 1.99 s to a Q3 of 3.01 s.
- forward, right-of; (b) 0.000000000987.
- 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
- The spur length such that 30% of the turkeys have a longer spur length is 23 mm.
- The proportion of turkeys with spur lengths between 15 and 25 mm is 0.811.
- The proportion of turkeys with spur lengths greater than 30 mm is 0.007.
- The spur length such that 10% of turkeys have a smaller spur length is 16 mm.
- The proportion of urkeys with a spur length less than 18 mm is 0.217.
- 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)