$```$

In $[]$: import pandas as pd

import matplotlib.pyplot as plt

data $=$ pd$.$read$\_$csv$("./$alleghenyCensusTractIncome$\_$processed.csv$")$

data $=$ data$[$data$["$Type$"]==$ "Households"$][["$Census Tract","Mean income $($dollars$)"]]$

print$("$Number of Census Tracts: $\%$d$"\%$ len$($data$))$

data$["$Mean income $($dollars$)"].$hist$()$

plt$.$xlabel$("$Avg$.$ Annual Household Income $($$$)",$fontsize$=15)$

plt$.$ylabel$("$Number of Census Tracts",fontsize$=15)$

data.head$()$

$```$

If I take a random sample of $50$ Census Tracts, what is the probability that the sample's expected value falls between $\backslash (\backslash$$ $100,000\backslash )$ and $\backslash (\backslash$$ $110,0000\backslash )?$

Previously, we answered this kind of question by referring to the following plot of the normal distribution. We will discuss how to calculate these probabilities for any interval on the distribution in a future module. But, for now, know that the area under the curve is representing the probability that the sample mean falls within an interval. For example, the probability that the sample mean is between the population mean and $1$ standard error above the mean $($i$.$e$.,$ between $0$ and $\backslash (1\backslash$sigma $\backslash )$ in the plot$)$ is $\backslash (34\mathrm{.}1\backslash \%\backslash ).$

For now, I want you to use the Scipy Python package to answer this question with an exact probability. Scipy is a Python package designed for scientific computing and it contains many useful functions for statistics and machine learning. The "scipy.stats.norm" is a Python class for representing the Normal Distribution given an expected value and the standard deviation. $```$

In $[]$: # Here is an example using scipy.stats.norm to calculate the probability that the sample mean is below $2$ given that the populatic

import scipy

popExpectedValue $=3,$

standardError $=1$

N $=$ scipy.stats.norm$($loc$=3,$scale$=1)$

p $=$ N$.$cdf$(2)$

print$("$Probability the expected value of the sample is below $2$ is $\%0\mathrm{.}3$f$"\%$ p$)$

$```$

Use the cell below to answer the question about Allegheny County Census Tracts: If I take a random sample of Census Tracts, what is the probability that the sample's expected value falls between $\backslash (\backslash$$ $100,000\backslash )$ and $\backslash (\backslash$$ $110,0000\backslash )?$

$```$

In $[]$: import scipy

def calculate$\_$income$\_$distribution$($data$)$:

# your code here

raise NotImplementedError

return #your code here.

calculate$\_$income$\_$distribution$($data$)$

$```$