Using the Central Limit Theorem

What this lecture covers

This problem applies the central limit theorem to a factory production scenario: daily gadget output is normal, i.i.d., and the questions involve totals over many days. It works through three parts: computing a tail probability for a fixed number of days, finding the largest number of days satisfying a probability constraint, and computing a probability involving a random first-crossing day.

It builds on the central limit theorem and continuity correction from the main lecture. After watching, you should be able to standardize a sum of i.i.d. normal variables, apply continuity correction, solve backward for a sample size given a target probability, and rewrite an event about a random index as an equivalent event about a fixed sum.

Key ideas

• Setup: daily gadget output X_i is normal with mean 5 and variance 9, i.i.d. across days; sums of these are analyzed using the CLT.
• Standardizing a sum: for n days, the sum has mean 5n and variance 9n; subtracting the mean and dividing by the standard deviation converts any probability question into one about a standard normal Z.
• Continuity correction: since the sum is a discrete-like quantity, using a half-unit adjustment (e.g., 439.5 instead of 440) gives a more accurate normal approximation.
• Solving backward for n: given a target probability bound (such as wanting a tail probability under 0.05), you can look up the corresponding normal quantile and solve the resulting inequality for the largest valid n.
• Random index trick: the event "the first day the total exceeds 1000 is day 220 or later" is equivalent to "the sum over the first 219 days is still at most 1000," turning a question about a random stopping day into an ordinary CLT calculation on a fixed sum.

Before you watch

• Review the central limit theorem and continuity correction from the main lecture on the CLT.
• Be comfortable standardizing sums of independent normal random variables and reading normal distribution tables.

Check your understanding

1. Why does using 439.5 instead of 440 (or 439) improve the normal approximation to a discrete sum?
2. How would you solve for the largest n satisfying a probability bound like P(sum >= 200 + 5n) <= 0.05?
3. Why is the event "the first day total exceeds 1000 occurs on day 220 or later" equivalent to "the sum through day 219 is at most 1000"?
4. How does the mean and variance of the sum change as the number of days n increases, and how does that affect the standardized variable?