Download MDM4U The Normal Distribution Test 2

MDM4U The Normal Distribution Test 2
TOTAL MARK
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1. What is, usually, the random variable in an exponential probability
distribution? [K2]
Answer: Waiting time between succesive events in Poisson process (a
process in which events occur continuously and independently at a
constant average rate).
2. What is the standard normal distribution? [K2]
Answer: A normal distribution with 𝜇 = 0 and 𝜎 = 1is called the
standard normal distribution.
3. What does a z-score indicate? [C2]
Answer: The z-score of a datum x is 𝑧 =
𝑥−𝜇
𝜎
. It indicates the number
of standard deviations an observation or datum is above or below the
mean. Thus, a positive z-score indicates a datum above the mean, while
a negative z-score indicates a datum below the mean.
4. Calculate 𝑃(𝑋 < 9) for a normal distribution with 𝜇 = 10 and 𝜎 = 1.
[K2]
Soluton:
𝑃(𝑋 < 9) = 𝑃 (𝑍 <
9−10
1
) = 𝑃(𝑍 < −1) = 0.1587.
5. What is a continuity correction? [C2]
Answer: A continuity correction is an adjustment that is made when a
discrete distribution is approximated by a continuous distribution.
6. Use the normal approximation to find 𝜇 and 𝜎 in a binomial
distribution with n = 1000 and p = 0.5. [A2]
Soluton:
𝜇 = 𝑛𝑝 = 500, 𝜎 = √𝑛𝑝𝑞 = √250 = 5√10
7. A really tough trivia quiz has 50 multiple-choice questions, each with
four possible answers. Use a normal approximation to calculate the
probability of randomly guessing the correct answers for 10 to 15
(inclusive) of the questions. [A3]
Soluton:
𝜇 = 𝑛𝑝 = 50(0.25) = 12.5 ,
𝜎 = √𝑛𝑝𝑞 = √50(0.25)(0.75) ≈ 3.062,
𝑃(9.5 < 𝑋 < 15.5) ≈ 𝑃 (
= 𝑃(
9.5 − 12.5
3.062
9.5 − 12.5
3.062
<𝑍<
<𝑍<
15.5 − 12.5
3.062
15.5 − 12.5
3.062
)
)
≈ 𝑃(−0.98 < 𝑍 < 0.98)
= 𝑃(𝑍 < 0.98) − 𝑃(𝑍 < −0.98)
= 0.8365 − 0.1635
= 0.8365 − 0.1635
= 0.673.
8. Suppose the commuting time from North Bay to Barrie varies
uniformly from 135 min to 180 min. What is the probability that the
drive will take lesss than 160 min? [A3]
Answer:
5
9
9. The life spans of a particular species of turtle are normally distributed
with a mean of 180 years and a standard deviation of 40 years. What is
the probability that one of these turtles, selected at random, will live
more than a century?
Answer: 97.5%
10. The masses of Statistics students are belived to normally distributed.
The masses (in kg) of random sample of 36 sytudents are: [T6]
62
67
66
70
70
74
63
67
67
70
71
75
63
68
67
70
71
75
64
68
67
71
71
77
64
68
68
71
71
77
65
68
68
71
72
78
a) Determine the mean and standard deviation of these data.
Answer: 𝜇 = 69.31, 𝜎 = 4.02
b) What is the probability that the mass of a randomly selected Statistics
student will be at least 70 kg?
Answer: 43.08%
c) Of 120 Statistics students, how many would you expect to have a
mass greater than 65 kg?
Answer: 103.
11. Given the pdf
2𝑥
if 𝑥 ∈ [0; 3]
f (x) = { 9
0 otherwise,
a) Sketch the graph of f (x).
b) Determine: P (2 < x < 3).
5
Answer: .
9
c) Determine: P (x = 3)
d) Determine: P (2 ≤ x ≤ 3).