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Two Column Step by Step Example of a Sampling Distribution for a Mean
A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5.
Using a random sample of 50 students, find the probability that the mean stress score for the 50 students is less than 2.65.
| Guidelines | Example |
|---|---|
| Plan: State what we need to know. | We are asked the probability that the mean stress score of the sample of 50 students is less than 2.65. |
| Model: Think about the assumptions and check the conditions. |
Randomization Condition: The problem states that the sample was random. Independence Assumption: It is reasonable to think that the stress scores of 50 randomly selected students will be independent of each other. (But there could be exceptions, such as , if they were all from the same class that was having a major exam the next day.) 10% Condition: I assume the student body is over 500 students, so 50 students is less than 10% of the population. Sample Size Condition: Since the distribution of the stress levels is uniform, a larger sample size is needed. I believe that my sample of 50 students seems large enough. |
| State the parameters and the sampling model. | Since the individual stress scores follow a uniform distribution, X ~ U(1,5) where a=1 and b=5 (See Continuous Random Variables for the uniform).
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| Plot: Make a picture. Sketch the model and shade the area we’re interested in. |
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Mechanics: Let
= the mean stress score for the 75 students. Then,
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A z-score will be used:
Look this up in the table or use a computer. P(z<-2.15) = 0 |
| Conclusion: Interpret your result in the proper context, and relate it to the original question. | The probability that the mean stress score for the 50 college students is less than 2.65 is about 0. |
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