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Defective Phones
Question #3
Intent of Question
The primary goals of this question were to assess a
student's ability to
(1) recognize binomial distribution scenarios and
calculate relevant binomial probabilities;
(2) calculate an expected value based on the
binomial distribution;
(3) perform a conditional probability calculation
from a discrete random variable; and
(4) perform a probability calculation involving
independent events using the multiplication
rule.
Scoring:
This question is scored in three sections.
Section 1 consists of parts (a) and (b),
section 2 consists of part (c), and
section 3 consists of part (d).
Each of the three sections is scored as
essentially correct (E), partially correct (P), or
incorrect (I).
Question #3
A smartphone manufacturer is concerned about
the proportion of defectives produced at a
certain plant which produces many
smartphones every day.
Historically, approximately 15% of phones
produced at this plant have been defective. As
part of their quality assurance testing, 4
smartphones are selected at random from a
day's production.
Part (a)
Let X represent the number of defective phones
in the sample of 4 phones. Complete the table
below for the probability distribution of X,
assuming the historic defective rate holds.
x
0
1
0.522
0.368
2
3
4
P(x)
0.001
Part (a) solution
The random variable X has binomial distribution with
n = 4 and p = 0.15. The probability that exactly 2 of the
4 selected phones are defective is:
𝑃 𝑋=2 =
4
2
0.152 .852 ≈ 0.098
The probability that exactly 3 of the 4 selected phones
are defective is:
𝑃 𝑋=3 =
4
3
0.153 .851 ≈ 0.011
The completed probability distribution of X is displayed
below.
x
0
1
2
3
4
0.522
0.368
0.098
0.011
0.001
P(x)
Section 1 - continued
Part (b)
Part (b) solution
What is the expected number
of defective phones in this
sample?
The expected number of
defective phones in this
sample is:
𝐸 𝑋 = 𝜇 = 𝑛𝑝 = 4 0.15
= 0.60 phones.
Section 1 is scored as follows:
Essentially correct (E) if both probabilities are filled in correctly in the table in part
(a) with supporting work AND the expected value is calculated correctly in part (b)
with supporting work.
Partially correct (P) if both probabilities are filled in correctly in the table in part
(a) with supporting work AND the expected value is not calculated correctly in part
(b)
OR
the probabilities in part (a) AND the expected value in part (b) are both calculated
using incorrect (but reasonable) values for the binomial parameters n and p
OR
the response does not recognize this as a binomial setting and so the probabilities
in part (a) are not calculated correctly AND the expected value in part (b) is
calculated correctly using the formula for the mean of a discrete random variable
and the incorrect probabilities given in part (a)
OR
the correct answers are given in both part (a) and part (b) but sufficient supporting
work is not given in part (a) or part (b) or both.
Section 1 scoring continued
Incorrect (I) if the response does not meet the
criteria for E or P, including if a correct answer is
given in only one of parts (a) and (b) and sufficient
supporting work for the correct answer is not given.
Notes:
The calculator commands, binompdf(4, 0.15, 2) and
binompdf(4, 0.15, 3), are not considered sufficient
supporting work in part (a), unless somewhere in
the response for parts (a) and (b) the value 0.15 is
labeled as p and the value 4 is labeled as n.
Section 2 Part (c)
Part c
What is the probability that all
four phones were defective
given that at least two
defectives were found in the
sample?
Part c- solution
The probability that all four
phones were defective given that
at least two were defective is:
𝑃 𝑋=4𝑋≥2
𝑃(𝑋 = 4 and 𝑋 ≥ 2)
=
𝑃(𝑋 ≥ 2)
𝑃(𝑋 = 4)
=
𝑃(𝑋 ≥ 2)
0.001
=
0.098 + 0.011 + 0.001
≈ 0.009
Section 2 scoring
Essentially correct (E) if the probability is computed correctly, with
work shown that includes correct numerical values for both the
numerator and denominator.
Partially correct (P) if the response includes a numerator and
denominator in calculating the conditional probability, with one
(numerator or denominator) correct in numerical value and the other
incorrect;
OR
if the correct answer is given, but no work is shown.
Incorrect (I) if the response does not meet the criteria for E or P.
Section 3
Part (d)
Suppose that each day the
phones are produced
independently of all other
days' phones. If a sample of
size 4 is taken every weekday
(5 total samples), what is the
probability that there are no
defective phones found in any
of the 5 samples taken?
Part (d) solution
Because the phones are
produced independently
across days, the probability
that none of the five days'
samples contains a defective
phone is 0.522 5 ≈ 0.039.
Section 3 scoring
Essentially correct (E) if the response correctly calculates the
probability AND shows sufficient work.
Partially correct (P) if the response reports the correct
probability but shows no work or does not show sufficient work
OR
if the response uses the multiplication rule involving five events
but does so incorrectly and/or with an incorrect probability of
no defectives in a given day.
Incorrect (I) if the response does not meet the criteria for E or P.
Scoring
4
Complete Response
All three sections essentially correct
3
Substantial Response
Two sections essentially correct and one section partially correct
2
Developing Response
Two sections essentially correct and one section incorrect
OR
One section essentially correct and one or two sections partially correct
OR
Three sections partially correct
1
Minimal Response
One section essentially correct and two sections incorrect
OR
Two sections partially correct and one section incorrect