Using P(X=n) in normal distribution questions.

Question

Firstly, I know you don't use $P(X=n)$ in normal distribution questions. But I have a question where I'm kind of baffled on how to write the probability.

A scientist noted that 36% of temperature measurements were 4°C lower than the average.

I want to say I'd write $P(X<μ-4) = 0.36$ But it just doesn't make sense to me. I would instinctively write $P(X=μ-4) = 0.36$ because I'm looking at values where temperatures are exactly 4°C lower and not at least 4°C lower than the average. So where am I going wrong in understanding this?

Answer 1

You are correct to question this. I think the fault is in the statement

A scientist noted that 36% of temperature measurements were 4°C lower than the average.

Imagine taking lots of temperature measurements and finding that $36\%$ were exactly 4°C. It wouldn’t happen, since temperature is a continuous variable. Perhaps it should state:

A scientist noted that 36% of temperature measurements were 4°C lower than the average, to the nearest whole degree.

Or

A scientist noted that 36% of temperature measurements were at least 4°C lower than the average.

In either of these cases I’m sure you can figure out what probability to use.