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Step-By-Step Example of the Sampling Distribution Model for Sums (Ex 7.3)
An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.
Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7500.
| Guidelines | Example |
|---|---|
| Plan: State what we need to know. | We are asked to find the probability that the sum of the 80 values is more than 7500. |
| Model: Think about the assumptions and check the conditions. |
Randomization Condition: The sample was stated to be random. Independence Assumption: It is reasonable to think 80 randomly selected values will be independent of each other. 10% Condition: I assume the population is over 700, so 80 is less than 10% of the population. |
| State the parameters and the sampling model. | Since the distribution is unknown, but n is large enough
where Σx = 7500 |
| Plot: Make a picture. Sketch the model and shade the area we’re interested in. |
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Mechanics:
Let = the sum of the scores. Then,
where n = 80. Find the resulting probability from a table of Normal probabilities, a computer program or a calculator. |
A z-score will be used:
Look this up in the table or use a computer. P(z<2.236) = 0.987, P(z>2.236) = 1-0.987 = 0.013 |
| Conclusion: Interpret your result in the proper context, and relate it to the original question. | The probability that the sum of the scores is greater than 7500 is about 0.013. A the probability of a sum score this high or higher is only 1.3% |
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