Sampling Distribution and the Central Limit Theorem

Sampling Distribution and the Central Limit Theorem. Practice questions to deepen understanding of the sampling distribution and the Central Limit Theorem. Online statistics practice with full solutions and step-by-step explanations.

Sampling distribution and CLT practice — the Central Limit Theorem, distribution of the sample mean, standard error. Advanced academic statistics.

100 questions

Question 1
2.00 pts

📊 Basic concepts:
A researcher wants to study the average height of all residents of Israel.

What is the population in this study?

Explanation:

💡 Detailed explanation:

Step 1: what is a population? 🔍

Everyday explanation:

🌍 Population = all the items we are interested in

It is like asking: "Whom exactly do I want to know about?"

If I am studying the height of Israelis -
all Israelis are the population!

Step 2: mathematical definition 🎯

Population:

The set of all the items or people about whom we want to draw conclusions

Denoted by the letter: N

Population parameters:
• Mean: μ (mu)
• Standard deviation: σ (sigma)

Correct answer: All residents of Israel

Question 2
10.00 pts

📊 Basic concepts:
A researcher wants to examine the average height of all residents of Israel.

What is the population?

Explanation:
NULL
Question 3
10.00 pts

📊 Basic concepts:
A researcher selected 200 students from all high-school students in the country to measure their average grade.

What is the sample?

Explanation:
NULL
Question 4
2.00 pts

📊 Basic concepts:
A researcher chose 200 students from all the high school students in the country in order to examine their average grades.

What is the sample in this study?

Explanation:

💡 Detailed explanation:

Step 1: what is a sample? 🔍

Everyday explanation:

🔬 Sample = a small group we chose from the population

It is like tasting a spoonful from the pot -
you do not have to eat the whole pot to know the flavor!

The sample represents the population

Step 2: mathematical definition 🎯

Sample:

A subset of the population that has been chosen for the study

Denoted by the letter: n

Sample statistics:
• Sample mean: x̄ (x-bar)
• Sample standard deviation: s

Correct answer: The 200 students chosen

Question 5
2.00 pts

📊 Parameter and statistic:
The mean of the entire population is denoted by μ.
The mean of the sample is denoted by x̄.

Which of the following is a parameter?

Explanation:

💡 Detailed explanation:

Step 1: the essential difference 🔍

Everyday explanation:

🔒 Parameter = a number describing the population
It is a fixed number (but usually unknown to us!)

📊 Statistic = a number describing the sample
It is a variable number (depends on which sample we chose)

Step 2: memory rule 🎯

Memory trick:

🇬🇷 Greek letter (μ, σ) = parameter (population)
🔤 Latin letter (x̄, s) = statistic (sample)

Example:
• μ = 170 cm (mean height of all Israelis) - parameter
• x̄ = 172 cm (mean height of 100 people we measured) - statistic

Correct answer: μ - the population mean

Question 6
10.00 pts

📊 Parameter and statistic:
The mean of the entire population is denoted μ.
The sample mean is denoted x̄.

Which is a parameter?

Explanation:
NULL
Question 7
10.00 pts

📊 Sampling distribution:
Suppose many samples of size n are drawn from a population, and the mean of each sample is calculated.

What is the sampling distribution of the mean?

Explanation:
NULL
Question 8
2.00 pts

📊 Sampling distribution:
Suppose many samples of size n are taken from the same population, and the mean of each sample is computed.

What is "the sampling distribution of the mean"?

Explanation:

💡 Detailed explanation:

Step 1: the central idea 🔍

Everyday explanation:

Imagine doing an experiment:

1️⃣ Take a sample of 50 people → compute mean height
2️⃣ Take another sample of 50 people → compute mean height
3️⃣ Repeat many times...

The sampling distribution = all those means together!

Step 2: mathematical definition 🎯

Sampling distribution:

The distribution of a statistic (like the mean) when all possible samples of size n are taken from the population

In simple words:
If we take infinitely many samples and compute the mean of each -
we obtain a new distribution of all those means!

Correct answer: The distribution of all possible sample means

Question 9
2.00 pts

📊 Properties of the sampling distribution:
If the population mean is μ = 100,
what is the expected value (mean) of the sampling distribution of the mean?

Explanation:

💡 Detailed explanation:

Step 1: the amazing property 🔍

Everyday explanation:

🎯 When taking many samples and computing a mean for each:

• Some means will be a bit above μ
• Some will be a bit below μ
• But on average - we get exactly μ!

The mean of the means = the population mean!

Step 2: formula and meaning 🎯

The first property of the sampling distribution:

E(X̄) = μ

Meaning:
The expected value of the sample mean equals the population mean

Why does this matter?
It means that the sample mean is an unbiased estimator for the population mean!

Correct answer: 100

Question 10
10.00 pts

📊 Properties of the sampling distribution:
If the population mean is μ = 100,
what is the mean of the sampling distribution of x̄?

Explanation:
NULL
Question 11
10.00 pts

📊 Standard error:
If the population standard deviation is σ = 20 and the sample size is n = 100,
what is the standard error of the mean?

Explanation:
NULL
Question 12
2.00 pts

📊 Standard error:
If the population standard deviation is σ = 20 and the sample size is n = 100,

What is the standard error of the sample mean?

Explanation:

💡 Detailed explanation:

Step 1: what is the standard error? 🔍

Everyday explanation:

📏 Standard error = how much the means of different samples are "spread" around μ

• If the standard error is small → the means are close to μ (precise!)
• If the standard error is large → the means are spread out (less precise)

The larger the sample, the smaller the standard error!

Step 2: calculation 📊

Standard error formula:

SE = σ / √n

Substitution:
SE = 20 / √100
SE = 20 / 10 = 2

Step 3: meaning 💡

What did we learn?

✅ Standard error = 2 means that sample means usually lie within about 2 units of μ

✅ A sample of 100 is 10 times more precise than a sample of 1!
(because √100 = 10)

Correct answer: 2

Question 13
2.00 pts

📊 Effect of sample size:
What happens to the standard error of the sample mean when the sample size is multiplied by 4?

Explanation:

💡 Detailed explanation:

Step 1: the connection with the square root 🔍

Everyday explanation:

🔑 The key is in the formula: SE = σ/√n

Note: n is under the square root!

Therefore:
• If n grows by a factor of 4 → √n grows by √4 = 2
• Then SE shrinks by a factor of 2

To halve the standard error, we must quadruple n!

Step 2: numerical example 📊

Before: n = 25, SE = σ/√25 = σ/5
After (n × 4): n = 100, SE = σ/√100 = σ/10

Comparison: σ/5 → σ/10 = decreased by a factor of 2!

Step 3: golden rule 🎯

Important rule to remember:

To shrink the standard error by a factor of k,
we need to multiply the sample size by k²

Shrink SE by 2 → multiply n by 4
Shrink SE by 3 → multiply n by 9
Shrink SE by 10 → multiply n by 100

Correct answer: The standard error decreases by a factor of 2

Question 14
10.00 pts

📊 Sample size:
What happens to the standard error of the sample mean when the sample size is multiplied by 4?

Explanation:
NULL
Question 15
10.00 pts

📊 Shape of the distribution:
In which case is the sampling distribution of the mean always normally distributed?

Explanation:
NULL
Question 16
2.00 pts

📊 distribution:
distribution the sample/sampling of mean normally ?

Explanation:

💡 :

of 1: different 🔍

when distribution the sample/sampling normally?

🎯 1: population normally
→ distribution the sample/sampling always normally (every/all n!)

🎯 2: population No normally
→ distribution the sample/sampling normally (only if n two )

Question 17
2.00 pts

📊 Notation:
If X̄ is the sample mean, μ the population mean, and σ the population standard deviation,

How is the sampling distribution of X̄ denoted?

Explanation:

💡 Detailed explanation:

Step 1: breaking down the notation 🔍

Notation explanation:

📝 X̄ ~ N(μ, σ²/n) means:

• X̄ is normally distributed
• The expected value is μ (population mean)
• The variance is σ²/n
• (and therefore the standard deviation is σ/√n)

Step 2: why each other answer is wrong 📊

✗ X̄ ~ N(μ, σ²) — forgot to divide by n
✗ X̄ ~ N(0, 1) — that is only after standardization!
✗ X̄ ~ N(μ, σ) — σ instead of σ²/n

Remember: in the normal distribution N(μ, σ²) the second parameter is the variance (σ²), not the standard deviation!

Step 3: summary of formulas 🎯

Sampling distribution of X̄:

X̄ ~ N(μ, σ²/n)

Or equivalently:
• Expected value: E(X̄) = μ
• Variance: Var(X̄) = σ²/n
• Standard deviation (standard error): SE = σ/√n

Correct answer: X̄ ~ N(μ, σ²/n)

Question 18
10.00 pts

📊 Notation:
If X̄ is the sample mean, μ is the population mean, and σ is the population standard deviation,

what is the distribution of X̄?

Explanation:
NULL
Question 19
10.00 pts

📊 Calculation:
In a certain population the mean is μ = 50 and the standard deviation is σ = 25. A sample of size n = 100 is drawn.

What is the standard error?

Explanation:
NULL
Question 20
2.00 pts

📊 there is:
population mean is/it is μ = 50 and standard deviation the standard (deviation) is/it is σ = 15.
n = 36 observations.

what is standard error of mean sample?

Explanation:

💡 :

of 1: data 🔍

data:

• the population mean: μ = 50 (No on/about standard error)
• standard deviation population: σ = 15
• size sample: n = 36

of 2: there is standard error 📊

Question 21
2.00 pts

📊 Central Limit Theorem (CLT):

What does the Central Limit Theorem state?

Explanation:

💡 Detailed explanation:

Step 1: the amazing idea 🔍

Everyday explanation:

🌟 The CLT is one of the most important theorems in statistics!

It states:
No matter what the original population looks like -
if we take large enough samples and compute means,
the distribution of the means will be normal!

This works even if the population is totally weird!

Step 2: formal statement 🎯

Central Limit Theorem:

If X₁, X₂, ..., Xₙ is a random sample from a population with expected value μ and finite variance σ²,

then as n → ∞:

(X̄ - μ) / (σ/√n) → N(0,1)

In practice: for n ≥ 30 the approximation is usually good enough

Correct answer: When the sample size is large, the distribution of the sample mean approaches a normal distribution

Question 22
10.00 pts

📊 Central Limit Theorem (CLT):

What does the Central Limit Theorem state?

Explanation:
NULL
Question 23
10.00 pts

📊 Using the CLT:
In which case is it not necessary to rely on the CLT to conclude that X̄ is normally distributed?

Explanation:
NULL
Question 24
2.00 pts

📊 -CLT:
No on/about central distribution X̄ normally?

Explanation:

Explanation: See the definition and relevant formula above.

Question 25
2.00 pts

📊 Minimum sample size:
What is the minimum sample size commonly accepted for using the Central Limit Theorem?

Explanation:

💡 Detailed explanation:

Step 1: rule of thumb 🔍

Everyday explanation:

📏 n ≥ 30 is the commonly accepted "rule of thumb"

Why exactly 30?
• It is a number that is large enough for most distributions
• But not too large for practical needs

This is an accepted estimate, not an exact law!

Step 2: when more or less is needed 📊

Required sample size by population shape:

• Symmetric population: n ≥ 15
• "Normal" population: n ≥ 30 (the standard rule)
• Highly skewed population: n ≥ 50+

Step 3: summary 🎯

To summarize:

✅ n ≥ 30 - the most common accepted rule

⚠️ Exceptions:
• Symmetric population - n ≈ 15-20 may suffice
• Highly skewed population - need n > 50
• Normal population - any n is enough!

Correct answer: n ≥ 30

Question 26
10.00 pts

📊 Minimum sample size:
What is the commonly accepted minimum sample size to apply the CLT?

Explanation:
NULL
Question 27
10.00 pts

📊 Standardisation:
If X̄ ~ N(80, 16) (where 16 is the variance),

what is the Z-score when X̄ = 84?

Explanation:
NULL
Question 28
2.00 pts

📊 :
if X̄ ~ N(80, 16) ( 16 is/it is variance),

what is Z X̄ = 84?

Explanation:

💡 :

of 1: data 🔍

data what:

• X̄ ~ N(80, 16)
• : μ = 80
• variance: σ² = 16 → standard deviation: σ = √16 = 4
• : X̄ = 84

of 2: 📊

Question 29
2.00 pts

📊 there is :
the population mean μ = 100, standard deviation σ = 20, size sample n = 100.

what is the -X̄ > 102?

(given: P(Z < 1) = 0.8413)

Explanation:

💡 :

of 1: there is standard error 🔍

data:
• μ = 100, σ = 20, n = 100

standard error:
SE = σ/√n = 20/√100 = 20/10 = 2

of 2: 📊

Question 30
10.00 pts

📊 Probability calculation:
The population mean is μ = 100, standard deviation σ = 20, sample size n = 100.

What is P(X̄ > 102)?

Explanation:
NULL
Question 31
10.00 pts

📊 Two-tailed probability:
X̄ is normally distributed with mean 50 and standard error 5.

What is the probability that X̄ will be between 45 and 55?

Explanation:
NULL
Question 32
2.00 pts

📊 Two-tailed probability:
X̄ is normally distributed with expected value 50 and standard error 5.

What is the probability that X̄ will be between 45 and 55?

(Given: P(Z < 1) = 0.8413)

Explanation:

💡 Detailed explanation:

Step 1: standardize both values 🔍

Data: μ = 50, SE = 5

Lower-bound standardization:
Z₁ = (45 - 50) / 5 = -5/5 = -1

Upper-bound standardization:
Z₂ = (55 - 50) / 5 = 5/5 = 1

Step 2: compute the probability 🎯

Calculation:

P(45 < X̄ < 55) = P(-1 < Z < 1)

P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)

By symmetry: P(Z < -1) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587

Therefore:
P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826

Step 3: 68-95-99.7 rule 💡

Empirical rule to remember:

• 68% of values lie within ±1 standard deviation
• 95% of values lie within ±2 standard deviations
• 99.7% of values lie within ±3 standard deviations

The result 0.6826 ≈ 68% confirms this!

Correct answer: 0.6826

Question 33
2.00 pts

📊 Using the table:
X̄ is normally distributed with expected value 200 and standard error 10.

What is the probability that X̄ < 180?

(Given: P(Z < 2) = 0.9772)

Explanation:

💡 Detailed explanation:

Step 1: standardize 🔍

Data: μ = 200, SE = 10

Standardization:
Z = (180 - 200) / 10 = -20/10 = -2

We are looking for: P(X̄ < 180) = P(Z < -2)

Step 2: using symmetry 📊

Use of symmetry:

P(Z < -2) = P(Z > 2) (by symmetry!)

P(Z > 2) = 1 - P(Z < 2) = 1 - 0.9772

P(Z < -2) = 0.0228

Step 3: trick to remember 💡

Symmetry rule:

P(Z < -a) = P(Z > a) = 1 - P(Z < a)

This works because the normal distribution is symmetric about 0!

Correct answer: 0.0228

Question 34
10.00 pts

📊 Using the table:
X̄ is normally distributed with mean 200 and standard error 10.

What is P(X̄ < 180)?

Explanation:
NULL
Question 35
10.00 pts

📊 Critical value:
Population mean μ = 500, standard deviation σ = 100, sample size n = 25.

What is the 95th percentile of the sampling distribution?

Explanation:
NULL
Question 36
2.00 pts

📊 :
the population mean μ = 500, standard deviation σ = 100, size sample n = 25.

what is x -P(X̄ > x) = 0.05?

(given: Z₀.₀₅ = 1.645)

Explanation:

💡 :

of 1: there is standard error 🔍

data:
• μ = 500, σ = 100, n = 25

standard error:
SE = σ/√n = 100/√25 = 100/5 = 20

of 2: the/of 📊

Question 37
2.00 pts

📊 Effect of sample size:
What happens to the probability P(|X̄ - μ| < k) when the sample size is increased?
(where k is some constant)

Explanation:

💡 Detailed explanation:

Step 1: the reasoning 🔍

Everyday explanation:

When n increases:
• Standard error SE = σ/√n decreases
• The distribution becomes narrower
• More values fall close to μ

Therefore the probability of being within a fixed range increases!

Step 2: mathematical proof 🎯

Why does the probability grow?

P(|X̄ - μ| < k) = P(-k/SE < Z < k/SE)

As n grows → SE shrinks → k/SE grows

A larger range around 0 = a larger probability!

Correct answer: The probability increases

Question 38
10.00 pts

📊 Effect of sample size:
What happens to the probability P(|X̄ − μ| < k) when the sample size increases?
(k is fixed)

Explanation:
NULL
Question 39
10.00 pts

📊 CLT for sums:
If X₁, X₂, …, Xₙ is a sample from a population with mean μ and variance σ²,
what is the distribution of S = X₁ + X₂ + … + Xₙ?

Explanation:
NULL
Question 40
2.00 pts

📊 CLT for the sum:
If X₁, X₂, ..., Xₙ is a sample from a population with expected value μ and variance σ²,
what are the expected value and variance of the sum S = X₁ + X₂ + ... + Xₙ?

Explanation:

💡 Detailed explanation:

Step 1: properties of the sum 🔍

Everyday explanation:

If each X has expected value μ and there are n observations...

🎯 Expected value of the sum:
We add μ n times → E(S) = nμ

🎯 Variance of the sum:
Variances add (when the variables are independent)
→ Var(S) = nσ²

Step 2: comparison with the mean 📊

Sum S = ΣXᵢ:
• E(S) = nμ
• Var(S) = nσ²

Mean X̄ = S/n:
• E(X̄) = μ
• Var(X̄) = σ²/n

Step 3: connection between them 🎯

Connection between sum and mean:

X̄ = S/n

E(X̄) = E(S)/n = nμ/n = μ ✓

Var(X̄) = Var(S)/n² = nσ²/n² = σ²/n ✓

Correct answer: E(S) = nμ, Var(S) = nσ²

Question 41
2.00 pts

📊 there is with :
is distributed with mean 75 standard deviation 10.
of 36 students .

what is of ?

Explanation:

💡 :

of 1: data 🔍

data:
• mean : μ = 75
• standard deviation: σ = 10
• students: n = 36

: E(S) = ?

of 2: there is 📊

Question 42
10.00 pts

📊 Calculation with sums:
Exam scores are distributed with mean 75 and standard deviation 10.
A class has 36 students.

What is the expected total score of the class?

Explanation:
NULL
Question 43
10.00 pts

📊 Standard deviation of the sum:
Exam scores are distributed with mean 75 and standard deviation 10.
A class has 36 students.

What is the standard deviation of the total score?

Explanation:
NULL
Question 44
2.00 pts

📊 Standard deviation of a sum:
Test scores are distributed with mean 75 and standard deviation 10.
The sum of scores of 36 randomly selected students.

What is the standard deviation of the sum?

Explanation:

💡 Detailed explanation:

Step 1: variance formula 🔍

Recall:

Var(S) = nσ²

Therefore:
SD(S) = √(nσ²) = σ√n

Step 2: calculation 📊

Formula: SD(S) = σ × √n

SD(S) = 10 × √36 = 10 × 6

SD(S) = 60

Step 3: comparison 🎯

Notice the difference:

• Standard deviation of a sum: SD(S) = σ√n = 10×6 = 60
• Standard error of a mean: SE = σ/√n = 10/6 = 1.67

The sum is far more "spread" than the mean!

Correct answer: 60

Question 45
2.00 pts

📊 Unbiased estimator:
Why is the sample mean X̄ considered an "unbiased estimator" for the population mean μ?

Explanation:

💡 Detailed explanation:

Step 1: what is an unbiased estimator? 🔍

Everyday explanation:

🎯 Unbiased estimator = an estimator that on average "hits the target"

This does not mean every sample yields exactly μ!
But on average over many samples - we get μ

There is no systematic "tilt" toward one side

Step 2: formal definition 🎯

Definition:

An estimator θ̂ is unbiased for parameter θ if:

E(θ̂) = θ

For the sample mean: E(X̄) = μ ✓
Therefore X̄ is an unbiased estimator for μ

Correct answer: Because E(X̄) = μ

Question 46
10.00 pts

📊 Unbiased estimator:
Why is the sample mean X̄ considered an "unbiased estimator" of the population mean μ?

Explanation:
NULL
Question 47
10.00 pts

📊 Consistency of an estimator:
What happens to the sample mean X̄ as the sample size n grows?

Explanation:
NULL
Question 48
2.00 pts

📊 Consistency of an estimator:
What happens to the sample mean X̄ as the sample size n tends to infinity?

Explanation:

💡 Detailed explanation:

Step 1: law of large numbers 🔍

Everyday explanation:

🎲 Imagine rolling a die:

• 10 rolls: mean may be far from 3.5
• 100 rolls: mean closer to 3.5
• 1000 rolls: mean very close to 3.5
• ∞ rolls: mean = exactly 3.5!

The more data, the closer the mean to μ!

Step 2: mathematical statement 🎯

Law of Large Numbers:

As n → ∞:

X̄ →ₚ μ

(X̄ converges in probability to μ)

An estimator with this property is called consistent

Correct answer: X̄ converges to the population mean μ

Question 49
2.00 pts

📊 CLT for a proportion:
In a survey of 400 people, 55% supported a certain proposal.
If the true population proportion is p = 0.5,

What is the standard error of the sample proportion p̂?

Explanation:

💡 Detailed explanation:

Step 1: standard error formula for a proportion 🔍

Formula:

Standard error of the sample proportion:

SE(p̂) = √[p(1-p)/n]

Step 2: calculation 📊

Data: p = 0.5, n = 400

Calculation:
SE = √[0.5 × 0.5 / 400]
SE = √[0.25 / 400]
SE = √0.000625

SE = 0.025

Step 3: meaning 🎯

What does this mean?

A standard error of 0.025 (or 2.5%) means:

• Sample proportions will usually fall within ±2.5% of p
• That is: usually between 47.5% and 52.5%
• The survey is fairly precise!

Correct answer: 0.025

Question 50
10.00 pts

📊 CLT for proportions:
In a survey of 400 people, 55% supported a proposal.
If the true population proportion is p = 0.5,

what is the standard error of the sample proportion?

Explanation:
NULL
Question 51
10.00 pts

📊 Weight problem:
Apple weights are normally distributed with mean 150 g and standard deviation 20 g.
A sample of 25 apples is drawn.

What is P(X̄ > 156.72)?

Explanation:
NULL
Question 52
2.00 pts

📊 problem of :
is distributed normally with mean 150 grams standard deviation 20 grams.
25 .

what is the mean on/about on/about 156 grams?

(given: P(Z < 1.5) = 0.9332)

Explanation:

💡 :

of 1: data there is SE 🔍

data:
• μ = 150 grams, σ = 20 grams, n = 25
• population normally → X̄ normal

standard error:
SE = σ/√n = 20/√25 = 20/5 = 4 grams

of 2: there is 📊

Question 53
2.00 pts

📊 problem of :
is distributed with mean 8 minutes standard deviation 3 minutes.
36 .

what is the mean will be between 7.5 -8.5 minutes?

(given: P(Z < 1) = 0.8413)

Explanation:

💡 :

of 1: there is standard error 🔍

data: μ = 8, σ = 3, n = 36

standard error:
SE = 3/√36 = 3/6 = 0.5 minutes

CLT (n=36≥30): X̄ is distributed normal

of 2: 📊

Question 54
10.00 pts

📊 Service time problem:
Bank service time is distributed with mean 8 minutes and standard deviation 3 minutes.
A sample of 36 customers is drawn.

What is P(7 < X̄ < 9)?

Explanation:
NULL
Question 55
10.00 pts

📊 Exam score problem:
Exam scores are normally distributed with mean 70 and standard deviation 15.
A sample of 9 students is drawn.

What is P(X̄ > 75)?

Explanation:
NULL
Question 56
2.00 pts

📊 problem of :
is distributed normally with mean 70 standard deviation 15.
9 students.

what is mean will be -65?

(given: P(Z < 1) = 0.8413)

Explanation:

💡 :

of 1: is/does/whether to use normally? 🔍

test/check:

• n = 9 (small/less -30!)
• ... population is distributed normally

✅ Yes X̄ is distributed normally !
(No CLT whenpopulation normally)

of 2: there is 📊

Question 57
2.00 pts

📊 Manufacturing problem:
A factory produces screws whose lengths are distributed with mean 5 cm and standard deviation 0.2 cm.
64 screws were chosen at random.

What is the probability that the mean length will be between 4.95 and 5.05 cm?

(given: P(Z < 2) = 0.9772)

Explanation:

💡 Detailed Explanation:

Step 1: Calculating the standard error 🔍

Data: μ = 5, σ = 0.2, n = 64

Standard error:
SE = 0.2/√64 = 0.2/8 = 0.025 cm

Step 2: Standardization 📊

Lower bound:
Z₁ = (4.95 - 5) / 0.025 = -0.05 / 0.025 = -2

Upper bound:
Z₂ = (5.05 - 5) / 0.025 = 0.05 / 0.025 = 2

Step 3: Probability calculation 🎯

Calculation:

P(-2 < Z < 2) = P(Z < 2) - P(Z < -2)

= 0.9772 - (1 - 0.9772)
= 0.9772 - 0.0228

= 0.9544

(this matches the 95% rule!)

Correct answer: 0.9544

Question 58
10.00 pts

📊 Manufacturing problem:
A factory produces bolts whose lengths are distributed with mean 5 cm and standard deviation 0.2 cm.
A sample of 64 bolts is drawn.

What is P(4.95 < X̄ < 5.05)?

Explanation:
NULL
Question 59
10.00 pts

📊 Income problem:
The average monthly income in the population is 12,000 with a standard deviation of 3,000.
A sample of 100 people is drawn.

What is the 95th percentile of the sampling distribution of X̄?

Explanation:
NULL
Question 60
2.00 pts

📊 Income problem:
The average monthly income in the population is $12,000 with standard deviation $3,000.
100 people were sampled.

Find the value x such that P(X̄ < x) = 0.95

(given: Z₀.₀₅ = 1.645)

Explanation:

💡 Detailed Explanation:

Step 1: Calculating the standard error 🔍

Data: μ = 12,000, σ = 3,000, n = 100

Standard error:
SE = 3000/√100 = 3000/10 = $300

Step 2: Finding the Z value 📊

We seek: P(X̄ < x) = 0.95

So P(Z < z) = 0.95

From the table: z = 1.645

Step 3: Calculating x 🎯

Inverting the standardization formula:

z = (x - μ) / SE

1.645 = (x - 12000) / 300

x - 12000 = 1.645 × 300 = 493.5

x = $12,493.5

Correct answer: $12,493.5

Question 61
2.00 pts

📊 problem of :
is distributed with mean 10 litres -100 " standard deviation 2 litres.
49 .

what is standard error of mean ?

Explanation:

💡 :

of 1: data 🔍

data:
• μ = 10 litres (No there is SE)
• σ = 2 litres
• n = 49

of 2: there is 📊

Question 62
10.00 pts

📊 Fuel consumption problem:
Fuel consumption per 100 km is distributed with mean 10 litres and standard deviation 2 litres.
A sample of 49 vehicles is drawn.

What is the standard error?

Explanation:
NULL
Question 63
10.00 pts

📊 Blood pressure problem:
Systolic blood pressure is normally distributed with mean 120 mmHg and standard deviation 15.
A sample of 25 people is drawn.

What is P(X̄ > 126)?

Explanation:
NULL
Question 64
2.00 pts

📊 Blood pressure problem:
Systolic blood pressure is normally distributed with mean 120 mmHg and standard deviation 15.
25 people were sampled.

What is the probability that the mean blood pressure will be above 126?

(given: P(Z < 2) = 0.9772)

Explanation:

💡 Detailed Explanation:

Step 1: Calculating SE 🔍

Data: μ = 120, σ = 15, n = 25
The population is normal → X̄ is normal

SE = 15/√25 = 15/5 = 3

Step 2: Standardization and calculation 📊

Standardization:
Z = (126 - 120) / 3 = 6/3 = 2

Probability:
P(X̄ > 126) = P(Z > 2) = 1 - P(Z < 2)
= 1 - 0.9772 = 0.0228

Correct answer: 0.0228

Question 65
2.00 pts

⚠️ :
there is standard error :
σ = 20, n = 100
SE = 20/100 = 0.2

what is the of?

Explanation:

💡 :

of 1: 🔍

❌ what is the :

SE = σ/n = 20/100 = 0.2

No Correct!

of 2: there is Correct 📊

✅ there is Correct:

SE = σ/√n = 20/√100 = 20/10 =
Question 66
10.00 pts

⚠️ Error identification:
A student calculated the standard error as follows:
σ = 20, n = 100
SE = 20/100 = 0.2

What is the error?

Explanation:
NULL
Question 67
10.00 pts

⚠️ Error identification:
Given: X̄ ~ N(50, 4).
A student said: "The standard error is 4."

Is this correct?

Explanation:
NULL
Question 68
2.00 pts

⚠️ :
given: X̄ ~ N(50, 4)
: "standard error is/it is 4"

is/does/whether is/it is ?

Explanation:

💡 :

of 1: 🔍

N(μ, σ²):

X ~ N(50, 4)
• metres (50) =
• metres (4) = variance (No standard deviation!)

Yes: σ² = 4 → σ = √4 = 2

of 2: 📊

<
Question 69
2.00 pts

⚠️ :
population is distributed (No normally).
sample of n = 10.
normal distribution -X̄.

is/does/whether ?

Explanation:

💡 :

of 1: 🔍

when X̄ is distributed normally?

1️⃣ population normally → always!
2️⃣ population No normally → only if n ≥ 30 (CLT)

of:
• population: (No normally) ❌
• sample: n = 10 (small/less -30) ❌

of 2: 📊

Question 70
10.00 pts

⚠️ Error identification:
The population has a uniform (non-normal) distribution.
A sample of n = 10 is drawn.
A student applies the normal distribution to X̄.

Is this justified?

Explanation:
NULL
Question 71
10.00 pts

📊 Lifespan problem:
The lifespan of a light bulb is distributed with mean 1000 hours and standard deviation 100 hours.
A sample of 64 bulbs is drawn.

What is the 95% interval for X̄?

Explanation:
NULL
Question 72
2.00 pts

📊 problem of :
of is distributed with mean 1000 hours standard deviation 100 hours.
64 .

what is range metres mean because 95% mean samples?

(given: Z₀.₀₂₅ = 1.96)

Explanation:

💡 :

of 1: there is SE 🔍

data: μ = 1000, σ = 100, n = 64

standard error:
SE = 100/√64 = 100/8 = 12.5

of 2: there is range 📊

95%: Z = ±1.96
Question 73
2.00 pts

📊 Comparing samples:
Two researchers sampled from the same population (σ = 30).
Researcher A sampled 100 people, researcher B sampled 400 people.

What is the ratio between their standard errors?

Explanation:

💡 Detailed Explanation:

Step 1: Calculating the standard errors 🔍

Researcher A (n = 100):
SE_a = 30/√100 = 30/10 = 3

Researcher B (n = 400):
SE_b = 30/√400 = 30/20 = 1.5

Step 2: Calculating the ratio 📊

Standard error ratio:SE_a / SE_b = 3 / 1.5 = 2A's standard error is 2 times larger!

Step 3: Explaining the relationship 🎯

Note:

• Sample B is 4 times larger than sample A
• But the standard error is only 2 times smaller

The reason: √4 = 2
To reduce SE by a factor of k, n must be increased by a factor of k²!

Correct answer: Researcher A's standard error is 2 times larger than B's

Question 74
10.00 pts

📊 Comparing samples:
Two researchers sampled from the same population (σ = 30).
Researcher A sampled 100 people; Researcher B sampled 400 people.

What is the ratio of their standard errors?

Explanation:
NULL
Question 75
10.00 pts

📊 Survey problem:
In a survey of 900 people, 60% supported the proposal.
If the true proportion is p = 0.58,

what is P(p̂ > 0.60)?

Explanation:
NULL
Question 76
2.00 pts

📊 problem of :
of 900 people, 60% .
if proportion is/it is p = 0.58,

what is the sample of 60% more?

(given: P(Z < 1.22) = 0.8888)

Explanation:

💡 :

of 1: there is SE for the proportion 🔍

data: p = 0.58, n = 900

standard error for the proportion:
SE = √[p(1-p)/n] = √[0.58×0.42/900]
= √[0.2436/900] = √0.000271
= 0.0164

of 2: 📊

Question 77
2.00 pts

📊 size sample :
standard error of mean 2.
the standard (deviation) of population is/it is σ = 20.

what is size sample ?

Explanation:

💡 :

of 1: 🔍

given: σ = 20, SE = 2 ()

what SE = σ/√n :

√n = σ/SE
n = (σ/SE)²

of 2: there is 📊

Question 78
10.00 pts

📊 Required sample size:
We want the standard error of the mean to be 2.
The population standard deviation is σ = 20.

What sample size is needed?

Explanation:
NULL
Question 79
10.00 pts

📊 Drug problem:
The action time of a drug is distributed with mean 4 hours and standard deviation 0.8 hours.
A sample of 16 observations is drawn. The population is normally distributed.

What is P(X̄ > 4.4)?

Explanation:
NULL
Question 80
2.00 pts

📊 problem of :
of is distributed with mean 4 hours standard deviation 0.8 hours.
16 . population is distributed normally.

what is the mean on/about on/about 4.3 hours?

(given: P(Z < 1.5) = 0.9332)

Explanation:

💡 :

of 1: there is SE 🔍

data: μ = 4, σ = 0.8, n = 16
population normally → X̄ normal (also -n=16)

standard error:
SE = 0.8/√16 = 0.8/4 = 0.2

of 2: there is 📊

Question 81
2.00 pts

📊 :
what No Correct distribution the sample/sampling of mean?

Explanation:

💡 :

every/all 🔍

. standard error always equal σ?
❌ No Correct! SE = σ/√n, No σ

. equal μ?
✓ Correct! E(X̄) = μ

. SE when-n increases?
✓ Correct! SE = σ/√n

Question 82
10.00 pts

📊 Combined concepts:
Which of the following is not true about the sampling distribution of the mean?

Explanation:
NULL
Question 83
10.00 pts

📊 Deep understanding:
Why is the Central Limit Theorem so important in statistics?

Explanation:
NULL
Question 84
2.00 pts

📊 In-depth understanding:
Why is the Central Limit Theorem so important in statistics?

Explanation:

💡 Detailed Explanation:

Step 1: Why is the CLT so important? 🔍

Everyday explanation:

🌟 In real life, most populations are not normal!

• Incomes - skewed right
• Waiting times - exponential
• Scores - not always symmetric

But thanks to the CLT, we can use
the tools of the normal distribution in any case!

This opens the door to all of modern statistics!

Step 2: Applications 📊

Thanks to the CLT we can:

✓ Build confidence intervals
✓ Perform hypothesis tests
✓ Estimate probabilities

Even when we know nothing about the shape of the population!

Correct answer: Because it allows us to use the normal distribution even when the population is not normal

Question 85
2.00 pts

📊 problem of :
is distributed with mean 12 minutes standard deviation 6 minutes.
-95% what, mean of 36 will be 14 minutes.

is/does/whether there is ?
(given: Z₀.₀₅ = 1.645)

Explanation:

💡 :

of 1: there is percentile -95 🔍

data: μ = 12, σ = 6, n = 36

standard error:
SE = 6/√36 = 6/6 = 1

percentile -95 of X̄:
x₀.₉₅ = μ + 1.645×SE = 12 + 1.645×1 = 13.645

of 2: there is 📊

Question 86
10.00 pts

📊 Bank problem:
Bank waiting time is distributed with mean 12 minutes and standard deviation 6 minutes.
For a sample of 36, does a mean of 14 minutes exceed the 95th percentile?

Explanation:
NULL
Question 87
10.00 pts

📊 Comparing distributions:
We have three distributions:
A. Distribution of the population X.
B. Distribution of one sample (data).
C. Sampling distribution of X̄.

Which has the smallest standard deviation?

Explanation:
NULL
Question 88
2.00 pts

📊 the/of :
there is of :
. distribution population X
. distribution sample (data )
. distribution the sample/sampling of X̄

No there is standard deviation more?

Explanation:

💡 :

of 1: the standard (deviation) 🔍

of standard:

• population: σ
• sample (data ): σ
• distribution the sample/sampling of X̄: σ/√n

because -n > 1, always: σ/√n < σ

of 2: 📊

Question 89
2.00 pts

📊 problem of :
is distributed with mean 50 " standard deviation 12 ".
144 .

what is the on/about on/about 7,344 "?

(given: P(Z < 2) = 0.9772)

Explanation:

💡 :

of 1: there is metres of 🔍

data: μ = 50, σ = 12, n = 144

:
E(S) = nμ = 144 × 50 = 7,200

standard deviation of :
SD(S) = σ√n = 12 × √144 = 12 × 12 = 144

of 2: there is 📊

Question 90
10.00 pts

📊 Electricity problem:
Daily electricity consumption is distributed with mean 50 kWh and standard deviation 12 kWh.
A sample of 144 days is drawn.

What is P(X̄ > 52)?

Explanation:
NULL
Question 91
10.00 pts

📊 Symmetry:
If P(X̄ > 105) = 0.1,
and the distribution of X̄ is symmetric about its mean,

what is P(X̄ < 95) if μ = 100?

Explanation:
NULL
Question 92
2.00 pts

📊 metres:
if P(X̄ > 105) = 0.1,
-distribution X̄ metres mean,

what is P(X̄ < 95) if μ = 100?

Explanation:

💡 :

of 1: metres 🔍

given: P(X̄ > 105) = 0.1

105 5 on/about μ = 100
95 5 -μ = 100

metres of distribution normally:
P(X̄ < 95) = P(X̄ > 105) = 0.1

of 2: 📊

Question 93
2.00 pts

📊 Sales problem:
Daily sales are distributed with mean $1,000 and standard deviation $200.
The mean of 25 sales days is examined.

What is the probability that the mean will be between $960 and $1,040?

(given: P(Z < 1) = 0.8413)

Explanation:

💡 Detailed Explanation:

Step 1: Calculating SE 🔍

Data: μ = 1000, σ = 200, n = 25

Standard error:
SE = 200/√25 = 200/5 = 40

Step 2: Standardization 📊

Lower bound:
Z₁ = (960 - 1000) / 40 = -40/40 = -1

Upper bound:
Z₂ = (1040 - 1000) / 40 = 40/40 = 1

Step 3: Probability calculation 🎯

Calculation:

P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)
= 0.8413 - 0.1587
= 0.6826

Correct answer: 0.6826

Question 94
10.00 pts

📊 Sales problem:
Daily sales are distributed with mean 1,000 and standard deviation 200.
For a sample of 100 days,

what is P(980 < X̄ < 1020)?

Explanation:
NULL
Question 95
10.00 pts

📊 Estimator properties:
An estimator is both unbiased and consistent.

What does this mean?

Explanation:
NULL
Question 96
2.00 pts

📊 estimator:
estimator is/it is also bias also consistent.

what is the ?

Explanation:

💡 :

of 1: 🔍

bias (Unbiased):
E(θ̂) = θ
mean, estimator " metres"

consistent (Consistent):
when-n → ∞, estimator metres
every/all there is more data, estimator two more

of 2: 📊

whenth
Question 97
2.00 pts

📊 problem of :
standard error of 5, when the standard (deviation) of population is/it is 50.
what every/all observation 10 $.

what is theon/about ?

Explanation:

💡 :

of 1: the/of size sample 🔍

data: σ = 50, SE = 5 ()

there is n:
n = (σ/SE)² = (50/5)² = 10² = 100

of 2: there is on/about 📊

Question 98
10.00 pts

📊 Research planning problem:
A researcher wants a standard error of 5, when the population standard deviation is 50.
Each observation costs 10.

What is the minimum cost to achieve the target?

Explanation:
NULL
Question 99
10.00 pts

📊 Summary question:
A population has μ = 80 and σ = 24.
A sample of n = 36 observations is drawn.

What is P(72 < X̄ < 88)?

Explanation:
NULL
Question 100
2.00 pts

📊 summary:
population with μ = 80 -σ = 24.
n = 36 observations.

what is mean sample will be range of ±8 the population mean?

(given: P(Z < 2) = 0.9772)

Explanation:

💡 :

of 1: there is SE 🔍

data: μ = 80, σ = 24, n = 36

standard error:
SE = 24/√36 = 24/6 = 4

of 2: range 📊

range :
μ ± 8 = 80 ± 8 = [72, 88]