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Significance of Difference of Proportions

Bike commuting is still uncommon, but Washington, DC, has a decent share. It has increased by over 1 percentage point in the last few years, but is this a statistically significant increase? In this exercise you will calculate the standard error of a proportion, then a two-sample Z-statistic of the proportions.

The formula for the standard error (SE) of a proportion is:

$$SE_P = \frac{1}{N}\sqrt{SE_n^2 - P^2SE_N^2}$$

The formula for the two-sample Z-statistic is:

$$Z = \frac{x_1 - x_2}{\sqrt{SE_{x_1}^2 + SE_{x_2}^2}}$$

The DataFrame dc is loaded. It has columns (shown in the console) with estimates (ending "_est") and margins of error (ending "_moe") for total workers and bike commuters.

The sqrt function has been imported from the numpy module.

This exercise is part of the course

Analyzing US Census Data in Python

View Course

Exercise instructions

  • Calculate bike_share by dividing the number of bikers by the total number of workers
  • Calculate the SE of the estimate of bikers and total workers, by dividing the MOE by Z_CRIT
  • Calculate the SE of the proportions: se_bike is the SE of the subpopulation \(SE_n\), bike_share is the proportion \(P\), and se_total is the SE of the population \(SE_N\)
  • Calculate \(Z\): \(x_1\) and \(x_2\) are the bike_share in 2017 and 2011; \(SE_{x_1}\) and \(SE_{x_2}\) are se_p in 2017 and 2011

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Set the critical Z score for 90% confidence
Z_CRIT = 1.645

# Calculate share of bike commuting
dc["bike_share"] = ____

# Calculate standard errors of the estimate from MOEs
dc["se_bike"] = ____
dc["se_total"] = ____
dc["se_p"] = sqrt(____**2 - ____**2 * ____**2)**0.5 / dc["total_est"]

# Calculate the two sample statistic between 2011 and 2017
Z = (dc[dc["year"] == 2017]["bike_share"] - ____) / \
    sqrt(____**2 + ____**2)
print(Z_CRIT < Z)
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