# Probability related to Normal Distribution

songoku
Homework Statement:
The distance walked by a person each day is assumed to be normally distributed with mean 12 km and standard deviation 0.19 km (for Monday to Friday) and mean 10 km and standard deviation 0.5 km for Saturday
a) In randomly chosen week, find the probability the person walks further on Saturday than on Friday
b) In randomly chosen week, find the probability that the mean distance walked by the person for the 6-day period is less than 11 km
Relevant Equations:
Normal Distribution

Linear Combination of Random Variable
a) Let X = distance walked on Friday and Y = distance walked on Saturday

X ~ N (12, 0.192) and Y ~ N (10, 0.52)

Let A = Y - X → A ~ N (-2 , 0.2861)

P(Y > X) = P(Y - X > 0) = P(A > 0) = 9.2 x 10-5

But the answer key is 0.026

Where is my mistake? Thanks

Delta2

Homework Helper
Your working looks correct. I wonder if there is a misprint, and the weekday SD should be 0.9 km? (0.19 looks suspiciously small.)

Delta2, songoku and FactChecker
songoku
Your working looks correct. I wonder if there is a misprint, and the weekday SD should be 0.9 km? (0.19 looks suspiciously small.)
I calculate using 0.9 km and I got same answer as the answer key.

How to know intuitively that 0.19 km is too small for standard deviation?

Thanks

FactChecker
Homework Helper
Gold Member
2022 Award
I calculate using 0.9 km and I got same answer as the answer key.

How to know intuitively that 0.19 km is too small for standard deviation?

Thanks
The 0.19 looks suspicious because it is a more complicated number than the 0.5. Why not just 0.2? Working backwards from the given answer arrives at 0.9.

songoku and FactChecker
songoku
The 0.19 looks suspicious because it is a more complicated number than the 0.5. Why not just 0.2? Working backwards from the given answer arrives at 0.9.
It maybe a misprint (like suggested by mjc123) and based on the question, the standard deviation would be 0.91, which I think is also a complicated number than 0.5

I thought mjc123 has something like more intuitive explanation, something related to why the number is too small compared to other data given by the question. I thought like this because on other thread (forget when I posted it), I miscalculated the standard deviation and other helper said my standard deviation is suspiciously too small / too big. I tried to find that thread to re-read the explanation but I couldn't find it.

Thank you very much for all the help and explanation

Homework Helper
Gold Member
2022 Award
the standard deviation would be 0.91, which I think is also a complicated number than 0.5
Yes, but that is an output number, not an input number.

songoku