To solve this, we need to use the binomial distribution, as each question is an independent Bernoulli trial with a success probability $p = 0.85$.

Let:

• $X$ be the number of questions answered correctly,
• $n = 100$ (total questions),
• $p = 0.85$ (probability of answering correctly).

We want to find the probability $P(77 \leq X \leq 87)$, which is the sum of the probabilities from $X = 77$ to $X = 87$.

Step 1: Approximate with a Normal Distribution

Since $n$ is large, we can use the normal approximation to the binomial distribution. According to the approximation, if $X \sim \text{Binomial}(n, p)$, then $X$ can be approximated by $Y \sim N(\mu, \sigma^2)$, where:

• $\mu = np = 100 \times 0.85 = 85$,
• $\sigma = \sqrt{np(1 - p)} = \sqrt{100 \times 0.85 \times 0.15} \approx 3.5693$.

Step 2: Apply the Continuity Correction

To account for the discrete nature of $X$, we use a continuity correction. We seek: $P(76.5 \leq Y \leq 87.5)$ where $Y \sim N(85, 3.5693^2)$.

Step 3: Standardize and Calculate

Standardize the values: $P\left( \frac{76.5 - 85}{3.5693} \leq Z \leq \frac{87.5 - 85}{3.5693} \right)$

1. For $76.5$: $\frac{76.5 - 85}{3.5693} \approx -2.38$
2. For $87.5$: $\frac{87.5 - 85}{3.5693} \approx 0.70$

Now, we find the probabilities using a standard normal table: $P(-2.38 \leq Z \leq 0.70) = P(Z \leq 0.70) - P(Z \leq -2.38)$

From standard normal tables:

• $P(Z \leq 0.70) \approx 0.7580$,
• $P(Z \leq -2.38) \approx 0.0087$.

Thus: $P(77 \leq X \leq 87) \approx 0.7580 - 0.0087 = 0.7493$

Final Answer

$P(77 \leq X \leq 87) \approx 0.7493$

Would you like further details or explanations?

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Here are some related questions for further exploration:

1. What is the probability of getting exactly 85 questions correct?
2. How would the probability change if the success rate was 90%?
3. How is the continuity correction applied in normal approximations?
4. Why is the binomial distribution approximated by the normal distribution?
5. How would we calculate $P(77 \leq X \leq 87)$ without normal approximation?

Tip: When using normal approximation for binomial distributions, remember to apply the continuity correction to improve accuracy.