To solve this problem using the z-test function, we are essentially looking to find the proportion of students whose scores fall within a specific range around the mean.

Here’s how to approach it:

Step 1: Identify the relevant parameters

We are given:

• The mean score, $\mu = 317$
• The standard deviation, $\sigma = 46$
• The range is 73 points from the mean, so the scores we are interested in are:
  • Lower bound: $317 - 73 = 244$
  • Upper bound: $317 + 73 = 390$

Step 2: Standardize the scores (convert to z-scores)

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

• $X$ is the raw score
• $\mu$ is the mean
• $\sigma$ is the standard deviation

We now compute the z-scores for both the lower bound (244) and the upper bound (390):

1. For $X = 244$: $z_1 = \frac{244 - 317}{46} = \frac{-73}{46} \approx -1.587$

2. For $X = 390$: $z_2 = \frac{390 - 317}{46} = \frac{73}{46} \approx 1.587$

Step 3: Find the proportion using the z-table or z-test function

Next, we need to find the probability that the z-score falls between $z_1$ and $z_2$. This is done by looking up the z-scores in the standard normal distribution table or using a z-test function.

• The cumulative probability for $z_1 = -1.587$ is approximately 0.113.
• The cumulative probability for $z_2 = 1.587$ is approximately 0.943.

The proportion of students whose scores are within 73 points of the mean is the difference between these two cumulative probabilities: $P(244 \leq X \leq 390) = P(z_2) - P(z_1) = 0.943 - 0.113 = 0.83$

Conclusion:

About 83% of students taking the exam receive a score that is within 73 points of the mean.

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Would you like more details or have any questions on this?

Here are 5 related questions you might find interesting:

1. How does the z-score relate to probability in other distributions?
2. What happens if the distribution isn’t normal—how would the answer change?
3. How can you compute the z-score for non-symmetric distributions?
4. Can you use the z-test to compare two different exam score distributions?
5. How does the concept of a standard error relate to the z-test?

Tip: Always check that the data follows a normal distribution before applying z-tests, as non-normal data can lead to inaccurate results.