# Probability related to Normal Distribution

• songoku
I understand it now :)In summary, the given conversation discusses the distances walked on Friday and Saturday and the probability of the Saturday distance being greater than the Friday distance. Using the given means and standard deviations, the probability is calculated to be 9.2 x 10^-5, but the given answer key suggests it should be 0.026. After considering the possibility of a misprint, recalculating with a standard deviation of 0.9 km results in the same answer as the answer key.

#### 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
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
mjc123 said:
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
songoku said:
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
haruspex said:
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

songoku said:
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
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 and Delta2
Thank you very much for the help mjc123 and haruspex

## 1. What is a normal distribution?

A normal distribution is a type of probability distribution that is commonly used to describe continuous random variables. It is characterized by a symmetrical bell-shaped curve, with the mean, median, and mode all being equal. Many natural phenomena, such as human height and IQ, follow a normal distribution.

## 2. How is the normal distribution related to probability?

The normal distribution is used to calculate the probabilities of events occurring within a certain range of values. The area under the curve represents the probability of a particular outcome. This makes it a useful tool for predicting the likelihood of various outcomes in a given situation.

## 3. What is the difference between standard normal distribution and normal distribution?

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. This makes it easier to calculate probabilities and compare different normal distributions. A normal distribution can have any mean and standard deviation, while the standard normal distribution has a fixed mean and standard deviation.

## 4. How is the normal distribution used in statistical analysis?

The normal distribution is used in statistical analysis to determine the likelihood of a particular outcome or event occurring. It is also used to calculate confidence intervals and perform hypothesis testing. In addition, many statistical models, such as linear regression, assume that the data follows a normal distribution.

## 5. Can any data set follow a normal distribution?

No, not all data sets follow a normal distribution. In order for a data set to be considered normally distributed, it must meet certain criteria, such as having a symmetrical bell-shaped curve and a mean, median, and mode that are equal. If a data set does not meet these criteria, it may follow a different type of distribution, such as a skewed distribution or a bimodal distribution.

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