From Kaplan:
Example: The z-test
When your company’s gizmo machine is working properly, the mean length of gizmos is 2.5 inches. However, from time to time the machine gets out of alignment and produces gizmos that are either too long or too short. When this happens, production is stopped and the machine is adjusted. To check the machine, the quality control department takes a gizmo sample each day. Today, a random sample of 49 gizmos showed a mean length of 2.49 inches. The population standard deviation is known to be 0.021 inches. Using a 5% significance level, determine if the machine should be shut down and adjusted.
From the answer section:
My question: Since the variance is known, why didn’t they find the Z-stat using z = observation - mean / standard deviation ?
If I draw a SINGLE observation as a sample, then yes, its standard deviation will be 0.021; however, we are dealing with the sample mean with 49 observations. The standard deviation of the sample mean will be the population standard deviation divided by the root of the number of observations.
What’s your opinion on clip-on ties?
Oh okay…from my Kaplan QuickSheet; “The Standard Error of the sample mean is the standard deviation of distribution of the sample means.”
The standard error only applies when the observation consists of n > 1? Whereas, when one “standardizes the observation” i.e. calculates the z score, the observation is a single observation.
This discussion is quite helpful to me.
Clip-on ties? LOL Let’s just say, you are talking to a man that ties either a four-in-hand knot or Prince Albert knot; it all depends on the width of the tie blade that I grab to appropriately pair with what I am wearing that day. Yes, I am a devoted follower of Alan Flusser.
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Think of the standard deviation as the standard error for a sample size of 1.
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