When are you supposed to use standard deviation and when are you supposed to use standard error for z-stats?
Below is an example. Thanks in advance
An airline is concerned about passengers arriving too late at the airport to allow for the additional security measures. The airline collects survey data from 1,000 passengers on their time from arrival at the airport to reaching the boarding gate. The sample mean is 1 hour and 20 minutes, with a sample standard deviation of 30 minutes, and the distribution of arrival times is approximately normal. Based on this sample, how long prior to a flight should a passenger arrive at the airport to have a 95% probability of making it to the gate on time?
A) Two hours, thirty minutes. B) One hour, fifty minutes. C) Two hours, ten minutes.
Your answer: C was correct!
For a normal distribution, 95% of the probability is less than +1.65 standard deviations from the mean. We use the right-hand tail of the distribution because we are not concerned with passengers arriving too early, only arriving too late.
To have a 95% probability of arriving on time, the passenger should allow one hour and twenty minutes + 1.65 × 30 minutes = 80 minutes + 49.5 minutes = 129.5 minutes = 2 hours, 9.5 minutes.
Gekko11, I can see how the above question’s framing would tempt many to calculate the standard error (using the sample st. deviation and square root of sample size, which are both given).
However, you are simply being asked to use your knowledge of the normal distribution’s basic properties, specifically that 95% of observations lie below 1.65 standard deviations above the mean. You are not given the actual population parameters and instead are being asked to base your conclusion on the sample statistics (the sample mean and st. deviation).
Please note that standard error is used when building confidence intervals for the population mean, i.e. constructing a symetrical interval around the sample mean (1 hour and 20 mins) which contains the true population mean at a specified degree of confidence (e.g. 95%).
However, I hope you see that this is not what you are being asked about in this question and thus you should not be attempting to use standard error in you answer.
hey thanks for your response, but just to clarify:
I should only use the standard error when solving for a CI?
I was reading some other forums where some were suggesting that when that when the population std. deviation is known, use std deviation otherwise just use the standard error.
I came across some question where the population std. dev. was provided but they still used the standard error to calculate the z-test. This is what threw me off.
Use standard error (st. dev/square root of sample size) when asked to build a confidence interval which, with a stated degree of confidence, contains the true population mean.
Sometimes the question will give you the st. deviation (or variance) of the actual population, in which case you should use that in the standard error computation (st. deviation of the population/squre root of sample size).
However, on many occasions the population st.deviation will not be provided and you will be given the st. deviation of the sample instead. If that is the case, use it to compute the standard error (st. deviation of the sample/square root of sample size).
In both cases you are calculating the standard error but using different standard deviations to do so.
If we were interested in passengers arriving too early AND too late then we would be looking at two sides of the mean and we would use 1.96 as you stated,We use the right hand tail of the distribution because we are NOT concerned with passengers arriving too early,only arriving too late.
