Suppose that the scores on a reading ability test are normally distributed with a mean of 65 and a standard deviation of 9. What proportion of individuals score at least 64 points on this test? Round your answer to at least four decimal places.

To find the proportion of individuals who score at least 64 points on the test, we can use the standard normal distribution (Z-distribution). The steps are as follows:

1. **Convert the raw score to a Z-score**:
   The Z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X \) is the raw score (64 in this case),
   - \( \mu \) is the mean (65),
   - \( \sigma \) is the standard deviation (9).

   Substituting the values:
   \[
   Z = \frac{64 - 65}{9} = \frac{-1}{9} \approx -0.1111
   \]

2. **Find the cumulative probability** for the Z-score:
   The cumulative probability \( P(Z \leq -0.1111) \) can be found using standard normal distribution tables or a calculator. The result gives the probability that a score is less than 64.

3. **Calculate the proportion of individuals scoring at least 64**:
   Since we need the proportion of individuals scoring **at least** 64, we take:
   \[
   P(X \geq 64) = 1 - P(Z \leq -0.1111)
   \]

   Using a Z-table or calculator, the cumulative probability for \( Z = -0.1111 \) is approximately 0.4550.

   Therefore:
   \[
   P(X \geq 64) = 1 - 0.4550 = 0.5450
   \]

So, the proportion of individuals who score at least 64 points on this test is approximately **0.5450**.

Would you like more details or have any questions about this solution?

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Here are 5 related questions:

1. What is the probability of scoring exactly 64 points on the test?
2. How would the proportion change if the mean were 70 instead of 65?
3. What proportion of individuals score below 50 points on the test?
4. What is the probability of scoring between 60 and 70 points?
5. How would you calculate the proportion for a non-normal distribution?

**Tip:** Knowing how to interpret and calculate Z-scores is essential for working with normally distributed data, especially when determining probabilities and proportions.

Learn how to calculate the proportion of individuals scoring at least 64 points on a reading ability test using the normal distribution and Z-score calculations. This problem involves understanding normal distribution, Z-scores, and cumulative probabilities.

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