To solve this problem, we need to calculate the probability that the average final exam grade of a sample of 35 students lies between 76 and 82. We are given the following information:

• Mean of the population ($\mu$) = 79
• Standard deviation of the population ($\sigma$) = 8.2
• Sample size ($n$) = 35

Step 1: Calculate the standard error (SE)

The standard error (SE) of the sample mean is calculated using the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

Substituting the values:

$SE = \frac{8.2}{\sqrt{35}} \approx \frac{8.2}{5.916} \approx 1.386$

Step 2: Convert the problem to a standard normal distribution (Z-scores)

Now, we need to convert the range 76 to 82 to Z-scores. The formula to compute a Z-score is:

$Z = \frac{X - \mu}{SE}$

For $X = 76$:

$Z_1 = \frac{76 - 79}{1.386} \approx \frac{-3}{1.386} \approx -2.17$

For $X = 82$:

$Z_2 = \frac{82 - 79}{1.386} \approx \frac{3}{1.386} \approx 2.17$

Step 3: Find the probabilities corresponding to these Z-scores

Using a Z-table or a standard normal distribution calculator:

• The probability of $Z = -2.17$ is approximately $P(Z < -2.17) \approx 0.0150$.
• The probability of $Z = 2.17$ is approximately $P(Z < 2.17) \approx 0.9850$.

Step 4: Calculate the probability between the two Z-scores

The probability that the sample mean is between 76 and 82 is the difference between these two probabilities:

$P(76 \leq \bar{X} \leq 82) = P(Z < 2.17) - P(Z < -2.17)$

$P(76 \leq \bar{X} \leq 82) = 0.9850 - 0.0150 = 0.9700$

Final Answer:

The probability that the average final exam grade of the sample is between 76 and 82 is approximately 0.9700.

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Would you like more details on how we calculated this, or do you have any questions? Here are some follow-up questions that could expand on this topic:

1. How does the sample size $n$ affect the standard error and, consequently, the probability?
2. What happens to the probability if the sample size were increased to 50?
3. What is the impact of a larger population standard deviation on the Z-scores?
4. How would the result change if the distribution of the population was normal instead of skewed?
5. What is the difference between population parameters and sample statistics in hypothesis testing?

Tip: As sample sizes increase, the sampling distribution of the sample mean tends to approximate a normal distribution more closely due to the Central Limit Theorem, even if the population distribution is skewed.