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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?

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 \( n = 100 \) and \( p = 0.04 \).
   - For the Poisson distribution, the parameter \( \lambda = n \times p = 100 \times 0.04 = 4 \).

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

3. **Calculate the Poisson probabilities** for \( X = 0, 1, 2, \dots, 15 \):
   - The PMF for a Poisson distribution is given by:
     \[
     P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
     \]

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 \( X = 0 \) to \( X = 15 \):

\[
\begin{array}{|c|c|c|c|}
\hline
X & \text{Using Binomial (a)} & \text{Using Poisson (b)} & \text{Difference (a - b)} \\
\hline
0 & 0.01687 & 0.01832 & -0.00145 \\
1 & 0.07029 & 0.07326 & -0.00297 \\
2 & 0.14498 & 0.14653 & -0.00155 \\
3 & 0.19733 & 0.19537 & 0.00197 \\
4 & 0.19939 & 0.19537 & 0.00402 \\
5 & 0.15951 & 0.15629 & 0.00322 \\
6 & 0.10523 & 0.10420 & 0.00104 \\
7 & 0.05888 & 0.05954 & -0.00066 \\
8 & 0.02852 & 0.02977 & -0.00125 \\
9 & 0.01215 & 0.01323 & -0.00108 \\
10 & 0.00461 & 0.00529 & -0.00069 \\
11 & 0.00157 & 0.00192 & -0.00035 \\
12 & 0.00049 & 0.00064 & -0.00016 \\
13 & 0.00014 & 0.00020 & -0.00006 \\
14 & 0.00004 & 0.00006 & -0.00002 \\
15 & 0.00001 & 0.00002 & -0.00001 \\
\hline
\end{array}
\]

### 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.

Learn how to calculate the minimum score required to achieve an A on an exam using the standard deviation and mean. This problem involves understanding the normal distribution and using the Z-score formula to determine the lowest score for an A.

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