Summary of the Video
The video opens at the General Motors Proving Ground in Milford, Michigan. We focus on testing prototype vehicles for compliance with regulations on pollutants in exhaust emissions. For the particular pollutant we look at, oxides of nitrogen (NOX), the amount emitted by engines of a specific type follows a normal distribution. GM estimates the mean and standard deviation of this distribution by testing prototype vehicles. We want to use this normal distribution to calculate the percentage of engines that have emissions above a stated limit.
We standardize observations by subtracting the mean and dividing by the standard deviation, as in the previous unit. Standardized observations follow the standard normal distribution, with mean 0 and standard deviation 1. The standard normal table gives for any standardized value z the decimal fraction of all observations from any normal distribution that are less than z when standardized.
GM statistician Tom Lorenzen explains that GM needs to know what percent of vehicles have NOX levels less than 1 gram per mile, the limit set by regulators. Standardize x = 1 to get a z -value, look in the standard normal table, and find 0.67 or 67%. Regulators allow as much as 40% above the limit. The 67% below the limit found here means 33% above. This is too close to the 40% allowed to risk going ahead with production of thousands of cars, so GM will redesign the engine.
Now we leave GM and visit Army basic training. The Army must outfit thousands of soldiers. Take helmets, for example. It's too expensive to stock rare sizes, so the 5% of soldiers with the biggest heads get custom helmets. How big is that? An Army study found that head circumferences vary normally with mean 22.8 inches and standard deviation 1.1 inches. First look at the standard normal table. The z with area 0.95 below it is about 1.65. So the top 5% in any normal distribution lie at or above 1.65 standard deviations above the mean. That's at or above 24.6 inches in the distribution of head sizes.