Central Limit Theorem Calculator on a TI-84 Calculator

Normal Approximation Using CLT:

\[ P(a \leq \bar{X} \leq b) \approx \text{normalcdf}\left(\frac{a - \mu}{\sigma/\sqrt{n}}, \frac{b - \mu}{\sigma/\sqrt{n}}\right) \]

Approximate Probability:

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1. What is the Central Limit Theorem?

The Central Limit Theorem (CLT) states that the sampling distribution of the mean will approach a normal distribution as the sample size increases, regardless of the population's distribution shape. This allows us to use normal approximation for probabilities involving sample means.

2. How to Use the TI-84 Calculator

On a TI-84 calculator, you can compute this using the normalcdf function:

\[ \text{normalcdf}((a-\mu)/(\sigma/\sqrt{n}), (b-\mu)/(\sigma/\sqrt{n})) \]

Steps:

Press [2ND] then [VARS] (DISTR)
Select 2:normalcdf()
Enter lower bound, upper bound, 0, 1
Press [ENTER] to calculate

3. When to Use This Approximation

Guidelines: The approximation works well when:

Sample size n ≥ 30 (for most distributions)
Population is not extremely skewed
For proportions, np ≥ 10 and n(1-p) ≥ 10

4. Calculator Instructions

Tips:

Enter population parameters (μ, σ)
Enter sample size (n)
Enter the desired range (a, b) for the sample mean
All values must be valid (σ > 0, n > 0)

5. Frequently Asked Questions (FAQ)

Q1: Why use normal approximation?
A: It simplifies calculations when the exact distribution is complex or unknown.

Q2: What if my sample size is small?
A: For n < 30, consider using t-distribution if population is normal.

Q3: How accurate is this approximation?
A: Accuracy improves with larger sample sizes and more symmetric populations.

Q4: Can I use this for proportions?
A: Yes, with μ = p and σ = √(p(1-p)), where p is the population proportion.

Q5: What if my population is already normal?
A: The sampling distribution will be exactly normal for any sample size.