Let's break down your question step by step.

Given Data:

The table shows the scores of 20 students in Mathematics in the 10th grade. We need to calculate two things:

1. Sample Proportion ‘p’ of students who scored more than 80%.
2. Standard Error of the Proportion given the population proportion $p = 0.20$.

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Part a: Sample Proportion ‘p’

Step 1: Identify how many students scored more than 80%

Let's list the scores greater than 80%:

• 87, 98, 93, 89, 90

Thus, there are 5 students who scored more than 80%.

Step 2: Sample Proportion ‘p’

The sample proportion $p$ is calculated as the number of students who scored more than 80% divided by the total number of students.

p = \frac{\text{Number of students who scored more than 80%}}{\text{Total number of students}}

There are 20 students in total, and 5 students scored more than 80%.

$p = \frac{5}{20} = 0.25$

So, the sample proportion of students who scored more than 80% is 0.25.

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Part b: Standard Error of the Proportion

The Standard Error (SE) of a proportion can be calculated using the formula:

$SE = \sqrt{\frac{p(1 - p)}{n}}$

Where:

• $p$ is the population proportion (given as 0.20),
• $n$ is the sample size (in this case, $n = 20$).

Step 1: Apply the formula

We are given the population proportion $p = 0.20$ and the sample size $n = 20$.

$SE = \sqrt{\frac{0.20(1 - 0.20)}{20}} = \sqrt{\frac{0.20(0.80)}{20}} = \sqrt{\frac{0.16}{20}} = \sqrt{0.008}$

$SE = 0.0894$

So, the standard error of the proportion is 0.0894.

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Summary of Results:

a. The sample proportion of students who scored more than 80% is 0.25.

b. The standard error of the proportion, given that the population proportion is 0.20, is 0.0894.

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Would you like further clarification on any of the steps or concepts? Here are some related questions:

1. How do you interpret the sample proportion in a real-world scenario?
2. How does changing the sample size affect the standard error?
3. What does the population proportion represent in this context?
4. How would you calculate the confidence interval for this proportion?
5. What might cause a discrepancy between the sample proportion and the population proportion?

Tip: When calculating the standard error, remember that it provides an estimate of how much the sample proportion is likely to fluctuate due to random sampling.