Q.Suppose that a 95% confidence interval states that population mean is greater than 100 and less than 300. How would you interpret this statement ?

Confidence Interval Interpretation

Imagine you're trying to guess the average height of all students in your school. You can't measure everyone, so you take a random sample of 100 students and find their average height is 165 cm. But you know that if you took a different random sample of 100 students, you'd probably get a slightly different average — maybe 164 cm, maybe 166 cm. The true school-wide average is somewhere around there, but you're not sure exactly where.

A confidence interval is a way of saying: "Based on my sample, I'm reasonably sure the true average lies somewhere between these two numbers." For example, you might calculate that the 95% confidence interval for the average height is (163 cm, 167 cm).

Why is it wrong? Because the true average is a fixed number — it either is between 163 and 167, or it isn't. There's no probability about it after you've calculated the interval. The probability was in the process before you collected the data.

The Correct Interpretation

Here's the precise statement: If you repeated this sampling process many, many times, and each time you calculated a 95% confidence interval, then about 95% of those intervals would contain the true population average.

Think of it like a fishing net. You cast your net (take a sample) and it lands somewhere in the ocean. The net covers a certain area (the confidence interval). The true fish (the population parameter) is somewhere in the ocean. You don't know if your particular net caught the fish. But you know that if you cast the net many times using the same method, about 95% of those casts would catch the fish.

A Concrete Example

Suppose you're a quality control manager at a factory that produces 1-litre bottles of juice. You take a random sample of 50 bottles, measure their volumes, and calculate a 95% confidence interval for the true average volume: (0.995 L, 1.005 L). …