To determine how many students scored between 70 and 80 points, we assume that the scores follow a normal distribution with a mean ($\mu$) of 70 and a standard deviation ($\sigma$) of 5.

Steps to Solve:

1. Convert the scores to z-scores: The z-score formula is: $z = \frac{x - \mu}{\sigma}$ For the lower bound $x = 70$: $z_1 = \frac{70 - 70}{5} = 0$ For the upper bound $x = 80$: $z_2 = \frac{80 - 70}{5} = 2$

2. Find the probabilities associated with these z-scores: Using the standard normal distribution table or a calculator:

   • For $z_1 = 0$: The cumulative probability is $P(Z \leq 0) = 0.5000$.
   • For $z_2 = 2$: The cumulative probability is $P(Z \leq 2) = 0.9772$.

3. Calculate the probability between the z-scores: The probability of scoring between 70 and 80 points is: $P(70 \leq X \leq 80) = P(Z \leq 2) - P(Z \leq 0)$ $P(70 \leq X \leq 80) = 0.9772 - 0.5000 = 0.4772$

4. Determine the number of students: Since the total number of students is 1000, the number of students scoring between 70 and 80 is: $\text{Number of students} = 0.4772 \times 1000 = 477.2$

Final Answer:

Approximately 477 students scored between 70 and 80 points.

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Related Questions:

1. What proportion of students scored below 70?
2. How many students scored above 80 points?
3. What is the z-score for a score of 85?
4. What percentage of students scored between 60 and 70 points?
5. How would the calculation change if the standard deviation were 10?

Tip:

When dealing with normal distributions, always sketch a quick bell curve and label the mean and standard deviations to visualize the problem better.