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### Course: AP®︎/College Statistics>Unit 11

Lesson 2: Setting up a test for a population mean

# Conditions for a t test about a mean

Example showing how to check which conditions have been met for a t test about a mean.

## Want to join the conversation?

• I don't get how you would do this type of problem where the standard deviation is from the population and not the sample:
"The amount of money collected each week by a city’s parking meter staff has been recorded for a decade and has an average of \$35,800 and a standard deviation of \$720. In February, a supervisor noticed that the weekly collection appeared to be smaller than usual. For the most recent 9 weeks, the average weekly collection was \$35,200. Is there significant evidence that someone among the meter staff has been skimming some of the collection into their own pockets? Assume that the data for the most recent 9 weeks can be viewed as a random sample and that the amount of money collected each week has a normal distribution. Test at level α = 0.05."
• Since we know the population standard deviation, we can perform a z-test.

First of all, the conditions are met:
9 weeks is less than 10% of a decade.
We can view the 9 weeks as a random sample.
The population is normally distributed.

𝑧 = (35200 − 35800)∕(720∕√9) = −2.5,
which gives us the probability of a random sample mean being \$35,200 or less as 𝑝 = 0.0062 < 𝛼 = 0.05,
meaning that the difference between the population mean and our sample mean is most likely not due to random chance, and we have significant evidence that someone is skimming off the top.

(Note that we immediately jump to the conclusion that someone is skimming, though the decrease in money could be due to other reasons.
Maybe the parking company lowered the fee, or there was a virus outbreak that caused people to stay home and not use public parking places as much.)