To determine the percentage of the class that scored between 70% and 100%, we use the properties of the normal distribution and the z-score formula:

Step 1: Z-Score Formula

The z-score formula is:
$z = \frac{x - \mu}{\sigma}$
Where:

• $x$: The raw score
• $\mu = 77\%$: The mean
• $\sigma = 8\%$: The standard deviation

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Step 2: Calculate the Z-Scores

We compute the z-scores for $x = 70\%$ and $x = 100\%$:

a) For $x = 70\%$:

$z = \frac{70 - 77}{8} = \frac{-7}{8} = -0.875$

b) For $x = 100\%$:

$z = \frac{100 - 77}{8} = \frac{23}{8} = 2.875$

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Step 3: Use the Z-Table or Normal Distribution Calculator

The z-score allows us to find the cumulative probability from the standard normal distribution table (or a calculator).

a) Probability for $z = -0.875$:

From the z-table:
$P(Z \leq -0.875) = 0.1908$
So, the probability above this z-score is:
$P(Z > -0.875) = 1 - 0.1908 = 0.8092$

b) Probability for $z = 2.875$:

From the z-table:
$P(Z \leq 2.875) = 0.9979$

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Step 4: Subtract the Probabilities

The percentage of the class scoring between $70\%$ and $100\%$ is the difference between the two probabilities:
$P(70 \leq X \leq 100) = P(Z \leq 2.875) - P(Z \leq -0.875)$
Substitute the values:
$P(70 \leq X \leq 100) = 0.9979 - 0.1908 = 0.8071$

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Final Answer:

The percentage of the class scoring between 70% and 100% is approximately 80.71%.

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Would you like further clarification or details? 😊

Related Questions:

1. How would the percentage change if the mean shifted to 80%?
2. What is the probability of scoring below 70%?
3. What z-scores correspond to the top 10% of the class?
4. How is the normal distribution applied in real-world exam scenarios?
5. How do you calculate percentages for scores between any two given limits?

Tip:

When using a z-table, always check if it provides cumulative probabilities from the left or areas above the z-score to avoid confusion!