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central limit theorem & confident interval

⭐ Central Limit Theorem (CLT)

The Central Limit Theorem says something surprisingly powerful:

If you take many random samples and compute their means,
the distribution of those sample means will be approximately normal,
even if the original population is not normal —
as long as the sample size is large enough.

📌 What the CLT guarantees

  • The shape of the sampling distribution of the mean becomes normal.
  • The center of that distribution is the true population mean \mu .
  • The spread of that distribution is
    \frac{\sigma}{\sqrt{n}}
    where
  • \sigma = population standard deviation
  • n = sample size

This spread is called the standard error.

🎯 Why it matters

The CLT is the reason we can:

  • use normal distributions to make inferences
  • build confidence intervals
  • run hypothesis tests
  • trust sample means as good estimators

Without the CLT, modern statistics basically collapses.

⭐ Confidence Intervals (CI)

A confidence interval gives a range of plausible values for a population parameter (usually the mean).

The general form is:

\text{estimate} \pm \text{margin of error}

For a mean, the formula is:

\bar{x} \pm z^* \cdot \frac{s}{\sqrt{n}}

Where:

  • \bar{x} = sample mean
  • s = sample standard deviation
  • n = sample size
  • z^* = critical value (e.g., 1.96 for 95% confidence)

📌 Interpretation

A 95% confidence interval means:

If we repeated the sampling process many times,
about 95% of the intervals we build would contain the true population mean.

It does not mean “there is a 95% chance the true mean is in this specific interval.”
The true mean is fixed; the interval is what varies.

⭐ How CLT and CI connect

Confidence intervals rely on the idea that:

  • The sampling distribution of the mean is normal
  • The standard error tells us how much sample means vary
See also  Types of random variables

Both of these come directly from the Central Limit Theorem.

So the logic is:

  1. CLT → sample means follow a normal distribution
  2. Normal distribution → we can use z‑scores
  3. z‑scores → we can build confidence intervals

This is why the CLT is the backbone of inferential statistics.

🎨 A simple analogy

Imagine you’re trying to guess the average height of all people in Norway.

  • Each sample you take gives a different mean.
  • Those sample means bounce around the true mean.
  • The CLT says those sample means form a bell curve.
  • A confidence interval is like saying:
    “Given how much sample means typically bounce around, here’s a range that probably contains the true average height.”

⭐ Examples of the Central Limit Theorem (CLT)

Example 1: Candy Weights

Suppose each piece of candy from a factory has a weight that is not normally distributed — maybe it’s skewed because machines sometimes overfill.

  • Population mean weight: 10 grams
  • Population standard deviation: 2 grams
  • Distribution: skewed right

Now take samples of size n = 40 and compute the sample mean each time.

Even though the individual weights are skewed, the distribution of sample means will be:

  • approximately normal
  • centered at 10 grams
  • with standard error
    \frac{2}{\sqrt{40}} = 0.316

This is the CLT in action:
skewed population → normal sampling distribution (if n is large).

Example 2: Rolling Dice

A single die roll is uniform, not normal.

But if you take the average of 30 rolls, and repeat that process many times, the distribution of those averages becomes:

  • bell‑shaped
  • centered at 3.5
  • with standard error
    \frac{\sigma}{\sqrt{30}}

Again: non‑normal population → normal sample means.

See also  geometric distribution is memoryless

⭐ Examples of Confidence Intervals (CI)

Example 1: Estimating Average Height

You measure the heights of 50 students.

  • Sample mean: \bar{x} = 170 cm
  • Sample standard deviation: s = 8 cm
  • Confidence level: 95% → z^* = 1.96

Standard error:
\frac{8}{\sqrt{50}} = 1.13

Margin of error:
1.96 \cdot 1.13 = 2.21

Confidence interval:
170 \pm 2.21 = (167.79,\ 172.21)

Interpretation:
We are 95% confident the true average height of the population is between 167.8 and 172.2 cm.

Example 2: Proportion of Students Who Own a Pet

Out of 200 students, 120 own a pet.

  • Sample proportion:
    \hat{p} = \frac{120}{200} = 0.60
  • Standard error:
    \sqrt{\frac{0.6(0.4)}{200}} = 0.0346
  • 95% CI:
    0.60 \pm 1.96(0.0346) 0.60 \pm 0.0678 (0.532,\ 0.668)

Interpretation:
We are 95% confident that 53.2% to 66.8% of all students own a pet.

⭐ A Combined Example (CLT + CI Together)

Estimating the Average Time Spent on TikTok

A researcher wants to estimate how long teenagers spend on TikTok daily.

They take a random sample of 64 teens.

  • Sample mean: \bar{x} = 95 minutes
  • Sample standard deviation: s = 30 minutes

Because of the Central Limit Theorem, the sampling distribution of the mean is approximately normal (n = 64 is large).

Standard error:
\frac{30}{\sqrt{64}} = \frac{30}{8} = 3.75

95% CI:
95 \pm 1.96(3.75)
95 \pm 7.35
(87.65,\ 102.35)

Interpretation:
We are 95% confident the true average TikTok time for teens is between 87.7 and 102.4 minutes per day.

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