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IITM BS Week 11 Statistics 2 Questions | Prasnya
Statistics 2 > Week 11
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Difficulty:
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All
7Q
01
Let
X
,
X
1
,
…
,
X
16
X, X_1, \dots, X_{16}
X
,
X
1
,
…
,
X
16
be i.i.d. samples from
N
(
μ
X
,
8
)
N(\mu_X, 8)
N
(
μ
X
,
8
)
and
Y
,
Y
1
,
…
,
Y
36
Y, Y_1, \dots, Y_{36}
Y
,
Y
1
,
…
,
Y
36
be i.i.d. samples from
N
(
μ
Y
,
18
)
N(\mu_Y, 18)
N
(
μ
Y
,
18
)
. The observed sample means are
X
ˉ
=
41
\bar{X} = 41
X
ˉ
=
41
and
Y
ˉ
=
40
\bar{Y} = 40
Y
ˉ
=
40
. We wish to test the hypotheses:
H
0
:
μ
X
=
μ
Y
H_0: \mu_X = \mu_Y
H
0
:
μ
X
=
μ
Y
against
H
1
:
μ
X
≠
μ
Y
H_1: \mu_X \ne \mu_Y
H
1
:
μ
X
=
μ
Y
. Consider a test that rejects
H
0
H_0
H
0
if
∣
X
ˉ
−
Y
ˉ
∣
>
c
|\bar{X} - \bar{Y}| > c
∣
X
ˉ
−
Y
ˉ
∣
>
c
. Let
α
\alpha
α
denote the significance level of the test and
F
Z
(
⋅
)
F_Z(\cdot)
F
Z
(
⋅
)
denote the cumulative distribution function (CDF) of the standard normal distribution. Express the critical value
c
c
c
in terms of
α
\alpha
α
and
F
Z
−
1
(
⋅
)
F_Z^{-1}(\cdot)
F
Z
−
1
(
⋅
)
. Select all the option(s) that apply.
Multiple correct
MEDIUM
3 marks
10 May 2026
02
Let
X
,
X
1
,
…
,
X
25
X, X_1, \dots, X_{25}
X
,
X
1
,
…
,
X
25
be i.i.d. samples from
N
(
μ
X
,
12
)
N(\mu_X, 12)
N
(
μ
X
,
12
)
and
Y
,
Y
1
,
…
,
Y
25
Y, Y_1, \dots, Y_{25}
Y
,
Y
1
,
…
,
Y
25
be i.i.d. samples from
N
(
μ
Y
,
24
)
N(\mu_Y, 24)
N
(
μ
Y
,
24
)
. The observed sample means are
X
ˉ
=
52
\bar{X} = 52
X
ˉ
=
52
and
Y
ˉ
=
49
\bar{Y} = 49
Y
ˉ
=
49
. We wish to test the hypotheses:
H
0
:
μ
X
=
μ
Y
H_0: \mu_X = \mu_Y
H
0
:
μ
X
=
μ
Y
against
H
1
:
μ
X
≠
μ
Y
H_1: \mu_X \ne \mu_Y
H
1
:
μ
X
=
μ
Y
. Consider a test that rejects
H
0
H_0
H
0
if
∣
X
ˉ
−
Y
ˉ
∣
>
c
|\bar{X} - \bar{Y}| > c
∣
X
ˉ
−
Y
ˉ
∣
>
c
. Let
α
\alpha
α
denote the significance level of the test and
F
Z
(
⋅
)
F_Z(\cdot)
F
Z
(
⋅
)
denote the cumulative distribution function (CDF) of the standard normal distribution. Express the critical value
c
c
c
in terms of
α
\alpha
α
and
F
Z
−
1
(
⋅
)
F_Z^{-1}(\cdot)
F
Z
−
1
(
⋅
)
. Select all the option(s) that apply.
Multiple correct
MEDIUM
3 marks
10 May 2026
03
Suppose
X
1
,
X
2
,
…
,
X
16
∼
i.i.d. Normal
(
μ
1
,
9
)
X_1, X_2, \dots, X_{16} \sim \text{i.i.d. Normal}(\mu_1, 9)
X
1
,
X
2
,
…
,
X
16
∼
i.i.d. Normal
(
μ
1
,
9
)
and
Y
1
,
Y
2
,
…
,
Y
8
∼
i.i.d. Normal
(
μ
2
,
25
)
Y_1, Y_2, \dots, Y_8 \sim \text{i.i.d. Normal}(\mu_2, 25)
Y
1
,
Y
2
,
…
,
Y
8
∼
i.i.d. Normal
(
μ
2
,
25
)
are independent samples. The sample means of these samples are denoted as
X
ˉ
\bar{X}
X
ˉ
and
Y
ˉ
\bar{Y}
Y
ˉ
respectively. Let the null and alternative hypothesis be
H
0
:
μ
1
=
μ
2
H_0: \mu_1 = \mu_2
H
0
:
μ
1
=
μ
2
and
H
A
:
μ
1
≠
μ
2
H_A: \mu_1 \neq \mu_2
H
A
:
μ
1
=
μ
2
. Choose the correct option(s) from the following.
Multiple correct
MEDIUM
3 marks
22 December 2024
04
Which hypothesis test should the researcher use to compare the effectiveness of the two teaching methods?
Comprehension
EASY
2 marks
01 September 2024
05
If the P-value of the test is 0.05, find the value of n. Round off your answer to the next greatest integer.
Comprehension
HARD
3 marks
01 September 2024
06
Consider two independent samples
X
1
,
X
2
,
…
,
X
50
∼
i.i.d. Normal
(
μ
1
,
25
)
X_1, X_2, \dots, X_{50} \sim \text{i.i.d. Normal}(\mu_1, 25)
X
1
,
X
2
,
…
,
X
50
∼
i.i.d. Normal
(
μ
1
,
25
)
and
Y
1
,
Y
2
,
…
,
Y
20
∼
i.i.d. Normal
(
μ
2
,
70
)
Y_1, Y_2, \dots, Y_{20} \sim \text{i.i.d. Normal}(\mu_2, 70)
Y
1
,
Y
2
,
…
,
Y
20
∼
i.i.d. Normal
(
μ
2
,
70
)
. Let the null and alternative hypothesis be:
H
0
:
μ
1
=
μ
2
H_0: \mu_1 = \mu_2
H
0
:
μ
1
=
μ
2
H
A
:
μ
1
≠
μ
2
H_A: \mu_1 \neq \mu_2
H
A
:
μ
1
=
μ
2
Suppose
T
=
Y
‾
−
X
‾
T = \overline{Y} - \overline{X}
T
=
Y
−
X
, where
Y
‾
=
Y
1
+
Y
2
+
⋯
+
Y
20
20
\overline{Y} = \frac{Y_1 + Y_2 + \dots + Y_{20}}{20}
Y
=
20
Y
1
+
Y
2
+
⋯
+
Y
20
and
X
‾
=
X
1
+
X
2
+
⋯
+
X
50
50
\overline{X} = \frac{X_1 + X_2 + \dots + X_{50}}{50}
X
=
50
X
1
+
X
2
+
⋯
+
X
50
. Consider a test that rejects
H
0
H_0
H
0
if
∣
T
∣
>
c
|T| > c
∣
T
∣
>
c
for some constant
c
c
c
. What is the size of the test in terms of 'c'?
Single correct
HARD
3 marks
30 April 2023
07
Suppose
X
1
,
X
2
,
…
,
X
15
X_1, X_2, \ldots, X_{15}
X
1
,
X
2
,
…
,
X
15
is an i.i.d. sample from a distribution Normal(
μ
1
,
4
2
\mu_1, 4^2
μ
1
,
4
2
). Suppose
Y
1
,
Y
2
,
…
,
Y
15
Y_1, Y_2, \ldots, Y_{15}
Y
1
,
Y
2
,
…
,
Y
15
is an i.i.d. sample from a distribution Normal(
μ
2
,
3
2
\mu_2, 3^2
μ
2
,
3
2
). Let
X
ˉ
\bar{X}
X
ˉ
and
Y
ˉ
\bar{Y}
Y
ˉ
be 40 and 43, respectively. Suppose we want to check if the distribution means are different. Find the P-value of the test. Enter the answer correct to two decimal places.
Comprehension
HARD
4 marks
07 August 2022
Showing 7 questions.
Complete question index for this section:
Question 1 (multiple):
Question 2 (multiple):
Question 3 (multiple):
Question 4 (comprehension):
Question 5 (comprehension):
Question 6 (single):
Question 7 (comprehension):
