Population proportion *statistics*

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SUMMARY

The discussion centers on calculating the approximate distribution of the sample proportion of Harley-Davidson motorcycle owners, which is 14% of the total registered motorcycles in the U.S. The sample size is 500, leading to the calculation of p-hat as 0.14. The conversation highlights the need to use normal probability calculations to determine the likelihood of the sample containing 20% or more Harley owners and at least 15%. The exact probability distribution is identified as a Binomial distribution, with a focus on how to apply the Normal distribution for approximation.

PREREQUISITES
  • Understanding of sampling distributions
  • Knowledge of Binomial and Normal distributions
  • Familiarity with statistical notation (p, p-hat, z-score)
  • Ability to perform probability calculations
NEXT STEPS
  • Learn about the Central Limit Theorem and its application in sampling distributions
  • Study the process of approximating Binomial distributions with Normal distributions
  • Explore the calculation of standard deviation for sample proportions
  • Review hypothesis testing for proportions in statistics
USEFUL FOR

Students studying statistics, educators teaching probability theory, and data analysts working with sample proportions in research.

georgeh
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Homework Statement



Harley-Davidson motorcyles make up 14% of all the motorcyles registered in the United States. You plan to interview an SRS of 500 motorcyle owners.
a) what the approximate distribution of the proportion of your sample who owns harleys?
b) Is your sample likely to contain 20% or more who own Harleys? Is it likely to contain at least 15% Harley owners? Do normal probability calculations to answer these questions

Homework Equations


p-phat = count of success in the sample/n
p = mean of the sampling distribution
z =( p-hat - p)/ std.dev


The Attempt at a Solution


I believe p = .14. n = 500
I don't understand what is meant by "approximate distribution of the proportion of your sample who owns harleys".
If they mean p-hat, would p-hat =(500*14/100) / 500 ? bit that would give me .14..
 
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Hint: The question says "Do normal probability calculations to answer these questions".

The "exact" probability distribution here is a Binomial distribution. How do you use a Normal distribution to approximate a Binomial distribution?
 

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