Thread: Help with Standard Normal Probabilities

1. Help with Standard Normal Probabilities

Hey,, so I need help with this problem, I'm not exactly sure how to start it either.

Suppose 75% of all families spend more than $75 weekly for food, while 15% spend more than$150. Assuming the distribution of family expenditures on groceries is normal, what is the mean weekly expenditure and standard deviation?

Thanks you guys!

2. Re: Help with Standard Normal Probabilities

Hey datpinecone.

You need to measure two quantities: mean and standard deviation. You also have two bits of data which means you can get a unique answer (which is what you want).

Basically you have P(X > 75) = 0.75 and P(X > 150) = 150 and X ~ N(mu,sigma^2).

Recall that the PDF of a normal is f(x;mu,sigma) = 1/SQRT(2*pi*sigma^2) * e^(-1/2*(x-mu)^2/sigma^2)) and P(X <= x) = Integral (-infinity,x) f(u;mu,sigma)du.

You are given two values of x with a corresponding P(X <= x) that have both the same mu and sigma.

You may need to use a computer to solve this problem and I'd recommend something like R or MATLAB/Octave.

3. Re: Help with Standard Normal Probabilities

Let's look at what we're given.

$\displaystyle \\\text{Let x=the amount of money a family spends on food weekly }\\Pr[X>75]=0.75\\Pr[X>150]=0.15\\\\\text{I prefer to rewrite them like this.}\\Pr[X<75]=0.25\\Pr[X<150]=0.85\\\\\text{We're trying to find }\mu \text{ and } \sigma\\\\\text{So, let's set this up using the standard normal distribution. Use a Normal table to find the Z for 0.25 and 0.85.}\\\\\frac{75-\mu}{\sigma}=-0.67\\\\\frac{150-\mu}{\sigma}=1.03\\\\\text{So, we have two equations and two unknowns. You should easily be able to find the two values. If you have more questions, post them up}$