MATH1725 Introduction to Statistics: Worked examples

Question (lecture 1-2).
For values 1, 3, 4, 5, 6 obtain the sample mean, sample median, sample variance and sample
standard deviation.
Answer: 1
Question (lecture 1-2).
The number of insurance policies sold by a small firm per week is 7, 8, 5, 6, 6, 7, 9, 5, 7, 8, 4, 7, 6,
7, 7, 5, 8, 6, 7, 6, 6. Obtain the sample mean, sample median, sample variance, sample standard
deviation. Check your values using R.
Answer: 2
Question (lecture 3).
For Z ∼ N(0,1), calculate pr{Z ≤ 0.55}, pr{Z > 2.25}, pr{Z ≤ −0.15}, pr{−1.50 < Z ≤ 2.25}.
Answer: 3
Question (lecture 3).
For Z ∼ N(0,1), calculate pr{Z ≤ 0.63}.
Answer: 4
Question (lecture 3).
For Z ∼ N(0,1), determine the value of z such that: pr{Z ≤ z} = 0.8944, pr{Z > −z} = 0.9713,
pr{−z < Z ≤ z} = 0.9108.
Answer: 5
Question (lecture 3).
An advertising company requires all of its job applicants to take a psychometric test. Based on
recent studies, it is believed that the test score follows a normal distribution with mean 100 and
standard deviation 15. Determine the probability that a job applicant will receive a test score
below 118, above 112, between 100 and 112.
Answer: 6
Question (lecture 4).
If X ∼ t 5 , for what value of x is pr{X > x} = 0.05
Answer: 7
Question (lecture 4).
If T ∼ t 8 , for what value t is pr{T > t} = 0.025 For what value t is pr{T ≤ t} = 0.05
Answer: 8
Question (lecture 4).
1 3.8, 4, 3.7, 1.92.
2 6.524, 7.0 (middle ordered value), 1.462, 1.209.
3 pr{Z ≤ 0.55} = Φ(0.55) = 0.7088, pr{Z > 2.25} = 1 − Φ(Z ≤ 2.25) = 1 − Φ(2.25) = 0.0122, pr{Z ≤ −0.15} =
1 − pr{Z ≤ 0.15} = 1 − Φ(0.15) = 0.4404, pr{−1.50 < Z ≤ 2.25} = pr{Z ≤ 2.25} − pr{Z ≤ −1.50} = 0.9210. Recall
that pr{Z > z} = 1 − pr{Z ≤ z}, pr{Z < −z} = pr{Z > z} by symmetry, and also pr{X < b} = pr{X < a} +
pr{a < X < b}.
4 Using interpolation in the tables Φ(0.63) = 0.7356.
5 pr{Z ≤ 1.25} = 0.8944, pr{Z > −1.90} = pr{Z ≤ 1.90} = 0.9713, pr{−z < Z ≤ z} = Φ(z) − Φ(−z) = 2Φ(z) −
1 = 0.9108 so Φ(z) = 0.9554 and z = 1.70.
6 0.8849, 0.2119, 0.2881. Hint: If X ∼ N(µ, σ 2 ), then pr{X ≤ x} = Φ ` x−µ
´.
7 σ
From tables, x = 2.015.
8 t 8(2.5%) = 2.306. pr{T > 1.860} = 0.05 so pr{T ≤ −1.860} = 0.05 by symmetry. Thus t = −1.860.
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