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The problem you have uploaded relates to finding probabilities using the sampling distribution of the sample mean.

### Information provided:
- Population mean (\(\mu\)) = 76.5
- Population standard deviation (\(\sigma\)) = 6.4
- Sample size (\(n\)) = 100

We can use the **Central Limit Theorem** here because the sample size is large enough. The sampling distribution of the sample mean will be approximately normal, with:
- **Mean of the sampling distribution** = Population mean = 76.5
- **Standard deviation of the sampling distribution (Standard error)**:
  \[
  \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{6.4}{\sqrt{100}} = \frac{6.4}{10} = 0.64
  \]

### a. Find the probability that the sample mean is greater than 77.
We want to find \( P(\bar{x} > 77) \). To do this, we need to standardize the value of 77 using the Z-score formula:
\[
Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}
\]
Substituting the values:
\[
Z = \frac{77 - 76.5}{0.64} = \frac{0.5}{0.64} \approx 0.78125
\]

Next, we find the probability corresponding to this Z-score. Using a Z-table or a standard normal distribution calculator, we find the area to the left of \( Z = 0.78125 \), which is approximately 0.7823. Thus, the probability that the sample mean is greater than 77 is:
\[
P(\bar{x} > 77) = 1 - 0.7823 = 0.2177
\]

### b. Find the probability that the sample mean is between 68 and 77.
We need to find \( P(68 < \bar{x} < 77) \). This involves calculating two Z-scores:

1. For \( \bar{x} = 68 \):
   \[
   Z_1 = \frac{68 - 76.5}{0.64} = \frac{-8.5}{0.64} \approx -13.28125
   \]
   A Z-score this extreme is essentially 0, so \( P(Z < -13.28) \approx 0 \).

2. For \( \bar{x} = 77 \), we've already found the Z-score to be 0.78125, and the corresponding probability is 0.7823.

Thus, the probability that the sample mean is between 68 and 77 is approximately:
\[
P(68 < \bar{x} < 77) = 0.7823 - 0 \approx 0.7823
\]

### Summary of answers:
- (a) The probability that the sample mean is greater than 77 is **0.2177**.
- (b) The probability that the sample mean is between 68 and 77 is **0.7823**.

Would you like more details or explanations?

### Related Questions:
1. How does the Central Limit Theorem apply to sample means?
2. What is the Z-score, and why is it important in probability calculations?
3. How does increasing the sample size affect the standard error?
4. What is the difference between population standard deviation and sample standard deviation?
5. How would the probability change if the standard deviation were larger or smaller?

### Tip:
When dealing with probabilities and sample means, always check if the sample size is large enough for the Central Limit Theorem to apply, which allows us to use the normal distribution for calculations.

In an examination taken by a large number of students, the mean score was 76.5 and the standard deviation is 6.4. A random sample of 100 test papers is taken. 
a. Find the probability that the sample mean is greater than 77. 
b. Find the probability that the sample mean is between 68 and 77.

Find the probability that the sample mean is greater than a given value or between two values using the Central Limit Theorem. This problem involves calculating Z-scores, standard errors, and understanding normal distribution in statistics. Suitable for high school or undergraduate students.

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