Normal Distribution — Part 3

Normal Distribution — Part 3. Practice questions to deepen understanding of the normal distribution — part 3. Online statistics practice with full solutions and step-by-step explanations.

Normal distribution Part 3 practice — advanced questions on the normal distribution, inverse computations, statistical applications.

40 questions

Question 1

2.50 pts

🏆 10% :
normal mean 80 standard deviation 6.
find 10% more?
known : \(P(Z > 1.28) \approx 0.10\).

-87.7

-86.0

-88.5

-92.0

Explanation:

X -P(X>X₀)=0.10. : P(Z>1.28)≈0.10, Yes z≈1.28.

calculate:
\(X = \mu + z\sigma = 80 + 1.28\cdot6 = 80 + 7.68 \approx 87.7\).

: 10% – 88.

Question 2

2.50 pts

📉 5% :
normal mean 70 standard deviation 9.
below find 5% more?
known : \(P(Z < -1.645) \approx 0.05\).

-55.2

-60.0

-52.5

-65.2

Explanation:

5% : P(Z

calculate :
\(X = \mu + z\sigma = 70 + (-1.645)\cdot9 \approx 70 - 14.8 = 55.2\).

Question 3

2.50 pts

⚠️ :
-5% :
z=+1.645, X=70+1.645·9≈84.8 "5% ".
?

-z , because -5% because because (P(Z < -1.645)).

mean -9.

z=0 because "".

χ² No Z.

Explanation:

5% find , Yes z . because No Correct – because .

Question 4

2.50 pts

⏱️ :
() normal mean 0.40 standard deviation 0.06 .
probability -0.52 ?

P(Z > 2) ≈ 0.0228

P(Z < -2) ≈ 0.0228

P(Z > 1) ≈ 0.1587

P(Z < 0) = 0.5

Explanation:

large mean = .

\(z = \dfrac{0.52-0.40}{0.06} = \dfrac{0.12}{0.06} = 2\).

Yes: P(X>0.52)=P(Z>2)≈0.0228.

Question 5

2.50 pts

👕 :
normal mean 175 cm standard deviation 7 cm.
-95% ( ).
( ) -95%?

161–189 cm

168–182 cm

154–196 cm

170–180 cm

Explanation:

95% area ≈ μ±2σ.

175±2·7 ⇒ 175±14 ⇒ 161–189.

Question 6

2.50 pts

🎨 area between because:
between z₁, 0, -z₂ ( z₁<0

probability ?

P(z₁ < Z < z₂)

P(Z > z₂)

P(Z < z₁)

P(Z > 0)

Explanation:

find between z₁ between z₂, 0 , Yes \(P(z_1 < Z < z_2)\).

Question 7

2.50 pts

📊 area between because No :
given: \(P(Z > 0.5) = 0.3085\), \(P(Z > 1.2) = 0.1151\).
\(P(0.5 < Z < 1.2)\) ?

0.1934

0.4236

0.1151

0.3085

Explanation:

area between because calculate between :
\(P(0.5 < Z < 1.2) = P(Z > 0.5) - P(Z > 1.2) = 0.3085 - 0.1151 = 0.1934\).

Question 8

2.50 pts

🧮 X→Z→area:
: μ=70, σ=8. probability between 66 -82?

P(-0.5 < Z < 1.5) ≈ 0.6247

P(-1 < Z < 1) ≈ 0.68

P(Z > 1.5) ≈ 0.0668

P(Z < -0.5) ≈ 0.3085

Explanation:

calculate Z :

66: \(z_1 = \dfrac{66-70}{8} = -0.5\)
82: \(z_2 = \dfrac{82-70}{8} = 1.5\).

: P(Z>0.5)=0.3085, P(Z>1.5)=0.0668 ⇒ P(-0.5

P(-0.5-0.5)-P(Z>1.5)=P(Z<0.5)-P(Z>1.5)=0.6915-0.0668≈0.6247.

Question 9

2.50 pts

📐 40% :
normal Z. a>0 -40% area find between -a -a.
Correct?

P(|Z| < a) = 0.40 ⇒ P(Z > a) = 0.30

P(|Z| < a) = 0.40 ⇒ P(Z > a) = 0.20

P(|Z| < a) = 0.40 ⇒ P(Z > a) = 0.10

P(|Z| < a) = 0.40 ⇒ P(Z > a) = 0.40

Explanation:

40% = 0.40, Yes 0.60. : 0.30. Yes P(Z>a)=0.30.

Question 10

2.50 pts

📏 :
: P(Z > 0.52) ≈ 0.3015.
a -40% ?

a ≈ 0.52

a ≈ 1.0

a ≈ 0.84

a ≈ 1.28

Explanation:

a -P(Z>a)≈0.30. , z≈0.52 -0.30 Yes a≈0.52.

Question 11

2.50 pts

💼 :
( ") normal μ=15, σ=3.
probability between 12 -21 "?

0.8186

0.68

0.95

0.5

Explanation:

calculate Z:

12 ⇒ z₁=(12-15)/3=-1
21 ⇒ z₂=(21-15)/3=2.

P(-1

: P(Z>1)=0.1587, P(Z>2)=0.0228.

area between -1 -2 = 1 - [P(Z<-1)+P(Z>2)] = 1 - [P(Z>1)+P(Z>2)] = 1 - (0.1587+0.0228) ≈ 0.8185 ≈ 0.8186.

Question 12

2.50 pts

⚠️ "between" No "":
P(Z>-1) "between -1 -2".
?

he forgot to subtract the part to the right of 2, namely P(Z > 2)

he should have added P(Z > 2) instead of subtracting it

No calculate Z.

he should have taken only P(Z < −1)

Explanation:

P(Z>-1) -2. area "between" because .

Question 13

2.50 pts

🏫 because:
because : μ=70, σ=5. 80.
because : μ=75, σ=7. 86.
" because " more?

, because z more .

, because more -100.

both are exactly the same distance from the mean

cannot be compared

Explanation:

: z = (80-70)/5 = 2.

: z = (86-75)/7 ≈ 11/7 ≈ 1.57.

z more ⇒ more mean because ⇒ more.

Question 14

2.50 pts

⚠️ :
: "86 more -80, more".
?

equal mean because, No .

equal between .

because always more.

86 No normal.

Explanation:

No . because , -Z equal .

Question 15

2.50 pts

📚 " ":
normal distribution, : " find ?"
more ?

μ ± 2σ

μ ± σ

μ ± 0.5σ

μ ± 10σ

Explanation:

μ±σ because -68% , μ±2σ because -95% – " ".

Question 16

2.50 pts

📊 probability -:
probability -|Z| > 2 ?

≈ 0.0456

≈ 0.0228

≈ 0.9544

≈ 0.9772

Explanation:

|Z|>2 :

P(|Z|>2)=P(Z>2)+P(Z<-2)=2·P(Z>2)=2·0.0228=0.0456.

Question 17

2.50 pts

📏 z area -:
0.05 ( 5%), ?

0.025

0.05

0.95

0.975

Explanation:

0.05 ⇒ 0.025.

statistic 5% -.

Question 18

2.50 pts

📏 z 2.5%?
known : \(P(Z > 1.96) \approx 0.025\).
?

|Z| > 1.96 for the extreme 5% (2.5% on each side)

only Z > 1.96 is possible

P(Z < 1.96) = 0.025.

P(Z = 1.96) = 0.025.

Explanation:

0.025 -z≈1.96. -1.96. -5% .

Question 19

2.50 pts

🧠 because 1.96?
z≈1.96 statistic?

because 2.5% (" 5%).

because always mean distribution.

because equal standard deviation.

because No .

Explanation:

z≈1.96 5% -: P(|Z|>1.96)≈0.05.

Question 20

2.50 pts

🎓 percentile 90:
log normal: μ=100, σ=15.
percentile 90 ( 10% )?
: P(Z > 1.28) ≈ 0.10.

-119

-112

-128

-115

Explanation:

z≈1.28. Yes:
X≈μ+zσ = 100 + 1.28·15 ≈ 100 + 19.2 = 119.2 ⇒ 119.

Question 21

2.50 pts

🎨 percentile :
small :

probability ?

P(Z > z₀) probability ( 0.10 0.05).

P(Z < z₀) probability .

P(Z = z₀) large.

the tail has no meaning

Explanation:

small probability because (, percentile 90, 95 ).

Question 22

2.50 pts

📑 No :
: P(Z > 1.37) ≈ 0.0853.
probability -Z large -1.37?

≈ 0.0853

≈ 0.1587

≈ 0.9147

≈ 0.5

Explanation:

– , probability.

Question 23

2.50 pts

🔁 z, :
\(P(Z < 1.37)\) ?

≈ 0.9147

≈ 0.0853

≈ 0.5

≈ 0.1587

Explanation:

P(Z<1.37) = 1 - P(Z>1.37) = 1 - 0.0853 = 0.9147.

Question 24

2.50 pts

🌡️ :
normal μ=36.8°C -σ=0.3°C.
probability 37.4°C?

P(Z > 2) ≈ 0.0228

P(Z > 1) ≈ 0.1587

P(Z < -2) ≈ 0.0228

P(Z < 0) = 0.5

Explanation:

z = (37.4-36.8)/0.3 = 0.6/0.3 = 2.

37.4 ⇒ P(Z>2)≈0.0228.

Question 25

2.50 pts

📐 80% :
because a>0 -80% area find between -a -a.
area ?

20%

10%

2.5%

Explanation:

80% ⇒ 20% .

Question 26

2.50 pts

📏 – a:
20%, 10%.
: P(Z > 1.28) ≈ 0.10.
a ?

a ≈ 1.28

a ≈ 0.84

a ≈ 2

a ≈ 1

Explanation:

P(|Z|a)=0.10 ⇒ a≈1.28.

Question 27

2.50 pts

📘 mean:
normal distribution, always Correct P(X > μ) ?

0.5

0.68

depends on the standard deviation

cannot be determined

Explanation:

μ: area find mean , no correlation .

Question 28

2.50 pts

⚠️ ?
: z=1.0, : 0.1587.
?

P(Z > 1.0) – area .

P(Z < 1.0).

always P(Z < -1.0).

percentile -84.

Explanation:

area -z. Yes 0.1587 P(Z>1.0).

Question 29

2.50 pts

🔁 , :
P(Z > 1.0) = 0.1587, P(Z < 1.0)?

0.8413

0.1587

0.5

0.3413

Explanation:

Explanation: Apply the relevant theorem or definition.

Question 30

2.50 pts

🎨 because, mean:
– – standard deviation equal, mean :

Correct?

standard deviation , mean – .

large more.

always more.

cannot see differences from a graph

Explanation:

( standard deviation), – mean large more.

Question 31

2.50 pts

📏 percentile 73:
z -P(Z < z) ≈ 0.73.
Correct?

look up a value close to 0.27 in the right tail (P(Z > z))

-0.73 .

always z=1 because "large -0.5".

z=0 because 0.73 " -0.5".

Explanation:

P(Z>z), P(Zz). 0.73 , 0.27 . Yes 0.27.

Question 32

2.50 pts

📏 – z:
: P(Z > 0.61) ≈ 0.2709.
z percentile 73?

z ≈ 0.61

z ≈ 0.27

z ≈ 1

z ≈ 0

Explanation:

-0.27 z≈0.61 ⇒ P(Z<0.61)≈0.73.

Question 33

2.50 pts

🚩 :
(-1%) X normal.
Correct z ?

P(Z > z) ≈ 0.01 ⇒ z large -2.3 .

P(Z > z) ≈ 0.01 ⇒ z small -0.

P(Z > z) ≈ 0.01 ⇒ z always = 1.

between z between .

Explanation:

1% mean. , because 2.3–2.4 1%.

Question 34

2.50 pts

🧠 calculate " "?
" " 95% , ?

-5% ( ).

no conclusion can be drawn

he is necessarily a failing student

he is necessarily a genius

Explanation:

-95% 5% – No always "" "because" .

Question 35

2.50 pts

⚠️ "standard deviation = "?
: " 10, No more -10 mean".
No Correct ?

No ; . because more mean.

standard deviation always 0.

mean always equal .

normal distribution because .

Explanation:

standard deviation No because between μ-σ -μ+σ, No area.

Question 36

2.50 pts

📘 between – :
normal μ=72, σ=9. probability between 60 -90?

0.8389

0.68

0.95

0.5

Explanation:

calculate Z:

60 ⇒ z₁=(60-72)/9=-12/9≈-1.33
90 ⇒ z₂=(90-72)/9=18/9=2.

: P(Z>1.33)≈0.0918, P(Z>2)≈0.0228.

area between -1.33 -2 = 1 - [P(Z<-1.33)+P(Z>2)] = 1 - [P(Z>1.33)+P(Z>2)] = 1 - (0.0918+0.0228) ≈ 0.8854.

because , 0.84–0.89. : ≈0.8389 more.

Question 37

2.50 pts

🔄 :
probability X normal distribution?

-Z, Z, .

-Z, mean , .

standard deviation , .

– .

Explanation:

large always: 1) X -Z. 2) ( 68–95–99.7). 3) .

Question 38

2.50 pts

⚠️ -Z :
: "P(X > 85) = P(Z > 85)".
No ?

85 X, No Z; calculate Z -85 .

Z always equal -X.

No calculate P(X > 85).

85 always mean.

Explanation:

X , Z z-score. z=(X-μ)/σ.

Question 39

2.50 pts

🔄 Why convert back to X?
After computing a probability or Z-score, why is it sometimes very important to convert the answer back to X (score, height, salary, etc.)?

because , cm " – No -Z.

because Z .

because X always equal -0.

because Z No .

Explanation:

z-score , (" ?") X.

Question 40

2.50 pts

✅ normal distribution -Z:
more ?

normal distribution μ -σ, z-score Z Z X area below – .

normal distribution .

z-score Z X No .

Z .

Explanation:

No: μ, σ, Z, Z – , , .