Translating the Population Data into Sample Data
A population is large. Think about the state of Texas. It is a large state. If we wanted to understand data in Texas, the entire population would be a ton of people! So, instead we take samples. These samples will disperse differently than a whole population, because they are smaller groups. Let's discuss how to determine that dispersion.
Translating the Population Data into Sample Data
*Note: this guide may use symbols you aren’t familiar with. View our complete list of Statistics Symbols here.
We begin by looking at an entire population to compare the mean and standard deviation. However, many times a problem will be further researched with a sample.
Recall: A sample is a small random group from the population.
Once you have mastered how the methods work with the population, you also need to master how they work with the sample. This involves an extra step added when tackling Real World Samples
(The mean is considered the same in the population and sample)
In a problem about averages/means, the standard deviation must be adjusted with this standard error formula above. The Standard Error is the standard deviation of the sampling distribution, adjusted to account for more potential error in a small sample versus the population.
The mean in a proportion is considered the average population proportion for the sample.
In a problem about proportions/percentages, the standard deviation must be adjusted with this standard error formula above.
This Standard Error is the standard deviation of the sampling distribution for a proportion, also adjusted to account for more potential error in a sample versus a population.
Now, you try! Practice Problems
Example One
Taco-eating contests have been found to be random normal distributions. Clarence wants to enter the contest, but is unsure if the taco-eating skills are enough to win. In researching the contest, Clarence finds that the record for 50 tacos eaten in the contest is 4.5 minutes. Clarence eats an average of 50 tacos in 5 minutes, with a standard deviation of 1.1 minutes. Clarence takes a sample of 16 practice contests. What are these values for the sample?
- What is the value of μx̅?
- What is the value of σx̅?
- Check at the end to be sure you understand the concept!
Example Two
Astronauts are sometimes called upon to serve longer trips than designated due to many factors. On average 15% of astronauts (12 out of every 80) serve longer than their mission date in space. An astronaut is curious if he will serve longer than his time. He gathers sample data from the 20 most recent trips.
- What is the value of μp̂?
- What is the value of σp̂?
Example One Solution
What is the value of μx̅?
What is the value of σx̅?
Example Two Solution
What is the value of μp̂?
What is the value of σp̂?
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