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

 

[mjc123]

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.)

 

[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

 

[haruspex]

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]

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

 

[haruspex]

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.

 

[mjc123]

I just thought 0.19 looked suspiciously small compared to 0.5, when the distances were similar. (Unless they were walking a much more well-defined route on weekdays, which I doubted.) So when your correct calculations were different from the book answer, I thought it worth seeing what I got if the SD was 0.9.

 

[songoku]

Thank you very much for the help mjc123 and haruspex