Math Problem Statement

On a certain exam, the mean is 72 marks, and the standard deviation is 9. If a grade of A is given to any student who scores at least two standard deviations above the mean, what is the lowest score that a person could receive and still get an A?

Solution

To solve the problem involving the binomial and Poisson distributions as depicted in the uploaded image, we will follow these steps:

  1. Determine the parameters:

    • The binomial distribution parameters are n=100n = 100 and p=0.04p = 0.04.
    • For the Poisson distribution, the parameter λ=n×p=100×0.04=4\lambda = n \times p = 100 \times 0.04 = 4.
  2. Calculate the binomial probabilities for X=0,1,2,,15X = 0, 1, 2, \dots, 15:

    • The probability mass function (PMF) for a binomial distribution is given by: P(X=k)=(nk)pk(1p)nkP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
  3. Calculate the Poisson probabilities for X=0,1,2,,15X = 0, 1, 2, \dots, 15:

    • The PMF for a Poisson distribution is given by: P(X=k)=λkeλk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
  4. Compute the difference between the binomial and Poisson probabilities.

  5. Fill in the table and comment on the results.

Let me calculate the values for you.Here are the computed probabilities for both the Binomial and Poisson distributions along with their differences for X=0X = 0 to X=15X = 15:

\hline X & \text{Using Binomial (a)} & \text{Using Poisson (b)} & \text{Difference (a - b)} \\ \hline 0 & 0.01687 & 0.01832 & -0.00145 \\ 1 & 0.07029 & 0.07326 & -0.00297 \\ 2 & 0.14498 & 0.14653 & -0.00155 \\ 3 & 0.19733 & 0.19537 & 0.00197 \\ 4 & 0.19939 & 0.19537 & 0.00402 \\ 5 & 0.15951 & 0.15629 & 0.00322 \\ 6 & 0.10523 & 0.10420 & 0.00104 \\ 7 & 0.05888 & 0.05954 & -0.00066 \\ 8 & 0.02852 & 0.02977 & -0.00125 \\ 9 & 0.01215 & 0.01323 & -0.00108 \\ 10 & 0.00461 & 0.00529 & -0.00069 \\ 11 & 0.00157 & 0.00192 & -0.00035 \\ 12 & 0.00049 & 0.00064 & -0.00016 \\ 13 & 0.00014 & 0.00020 & -0.00006 \\ 14 & 0.00004 & 0.00006 & -0.00002 \\ 15 & 0.00001 & 0.00002 & -0.00001 \\ \hline \end{array}$$ ### Comments on Results: - The Poisson approximation is close to the Binomial probability, especially for lower values of $$X$$, which is expected since $$n$$ is large and $$p$$ is small, making the Poisson approximation valid. - The differences are small but become more noticeable at higher $$X$$ values. Would you like more details on these results, or do you have any questions? Here are five related questions for further exploration: 1. How would the results change if $$p$$ were larger (e.g., $$p = 0.1$$)? 2. What is the impact on the differences when $$n$$ increases significantly? 3. How accurate is the Poisson approximation for different ranges of $$p$$? 4. Can we derive a general rule for when the Poisson approximation is appropriate? 5. How does the difference change for values of $$X$$ greater than 15? **Tip:** The Poisson distribution is often used as an approximation for the binomial distribution when $$n$$ is large and $$p$$ is small because the computational simplicity of the Poisson distribution can be beneficial in practical applications.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Standard Deviation

Formulas

Z-score formula: Z = (X - μ) / σ

Theorems

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Suitable Grade Level

High School