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Two Column Model Step by Step Example of a Sampling Distribution for a Binomial
Suppose in a local Kindergarten through 12 th grade (K-12) school district, 53 percent of the population favor a charter school for grades K-5. A simple random sample of 300 is surveyed.
Find the probability that at least 150 favor a charter school.
| Guidelines | Example from book |
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| Plan: State what we need to know. | We are asked the probability that number of people who favor a charter school is at least 150. |
| Model: Think about the assumptions and check the conditions. Check to see that these are Bernoulli trials. |
Randomization Condition: The sample was stated to be random. Independence Assumption: It is reasonable to think that the opinions of 300 randomly selected people are independent. 10% Condition: I assume the population of the school district is over 3000 people, so 300 people is less than 10% of the population. Success/Failure Condition success = favor charter school, failure = do not favor p = 0.53, (1 – p) = 0.47 np= 300(0.53) = 159>10, n(1-p)= 300(0.47) =141>10 |
| State the parameters and the sampling model. | Since this is a binomial distribution, X ~ B(300,0.53).
We will use the normal approximation to the binomial, Y~ N(159,8.6447) |
| Plot: Make a picture. Sketch the model and shade the area we’re interested in. |
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Mechanics: Since we want the probability of at least 150, 150 is included so we correct to 149.5. Then, we convert into a z score.
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A z-score will be used:
Look this up in the table or use a computer. P(z<-1.0989) = 0.136 P(z>-1.0989) = 1-0.136 = 0.864 |
| Conclusion: Interpret your result in the proper context, and relate it to the original question. | The probability that the number of people who favor charter schools is at least 150 is 86.4%. |
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