Quiz 1 401 Spring LuValle
This quiz is untimed and is to be solved either freehand or typed in a document (Please show
work! If you use a spreadsheet upload the spreadsheet as well).
Please keep track of the time it takes you to solve so I can calibrate timed tests. Submit the time
with the quiz. A pdf or picture may be uploaded to this assignment for me to grade
1) A certain protein is present in everyones saliva, but Covid 19 increases its concentration. An
instant test is developed based on this measuring this protein:
For people without covid 19, the protein has a concentration with a mean of 10 units/ml, and a
standard deviation of 2 units/ml. For people with covid 19 the protein has a concentration 21
units/ml and standard deviation of 1 unit/ml. Suppose we set a cutoff value of the test at 17
units/ml. So above 17 units/ml they are considered to have covid 19, and below 17 units they are
not.
A) Calculate under the assumptions given
a) What is the probability of falsely labelling a healthy person as having covid 19? *(show
work on all problems, e.g.screen shot of statkey, statcrunch, or hand calculations )
i)
Using Tchebychev’s inequality (upper bound)
1 Ez
K
t
3.5
I2g
1 3152 1
ii)
9184
1 0.08163
100 91.84
51.841
18.16 10.0816
Assuming both distributions are normal
positive
Lovid
Eithiitten
273.59
b) What is the probability of correctly saying a person does have covid 19 when they do?
(show work on all problems, e.g.screen shot of statkey, statcrunch, or hand calculations )
i)
Assuming both distributions are normal
2 Stone of 4
2
21
4
indicates almost
all values are inch
I
therefore
1210012
99.999687
B) Using normal distribution the of each problem calculate the probability of having Covid 19 if
the test says you have it if
a) The background proportion of people with active covid 19 is .03
A
B
Plac P CIA PCA
covidpositive
PCIA PCAtP IB P B
could negative
C persontestingpositive
0.0816 0.03
play 0.0816 0.037 10.9184
o.gr
b)The background proportion of people with active covid 19 is .5
999968 x s
P CIT
99977
C999968 x 5 C00023 5
0.00274041
2740411
offagged
percent
0.9926147
0.999770
2
PLC 0.5
PLN 1 0.5 0.5
C)Using normal distribution assumptions A calculate the probability of having Covid 19 if the
test says you dont have it if
a) The background proportion of people with active covid 19 is .03
PHD
a
p
neg
P CIL
tt
4 É cnn.pe cof.aaaa.aa
0.99994731
0.03Livid
P N 97 nowvid
P
c
b)The background proportion of people with active covid 19 is .5
PLN 1 0.5
5
co.gg 5 t
co
o.aaaT
1
2) Suppose the rate of meeting a bear at Jockey Hollow park in the fall is .004 per mile hiked.
Assuming a Poisson distribution, If you hike 12 miles in Jockey Hollow one day,
i)
what is the probability of meeting no bears
P
x0
ii)
e
y
IFI
1 951
What is the probability of meeting at least one bear.
1 9528 9.0472
I
i
athenifetitiing miles