In the question shown in the image, we are dealing with a bell-shaped (normal) distribution for standardized math test scores. The mean score is 530, and the standard deviation is 114. Let's analyze the three parts of the question:

(a) What percentage of standardized test scores is between 188 and 872?

We are asked for the percentage of scores falling between 188 and 872 in a normal distribution with a mean of 530 and a standard deviation of 114.

Step 1: Convert the raw scores to Z-scores using the formula:

$Z = \frac{X - \mu}{\sigma}$ Where:

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

For $X = 188$: $Z = \frac{188 - 530}{114} \approx -3.00$

For $X = 872$: $Z = \frac{872 - 530}{114} \approx 3.00$

Step 2: Use the standard normal distribution table (Z-table) or calculator.

The Z-table gives the area to the left of a Z-score:

• For $Z = -3.00$, the area is approximately 0.0013.
• For $Z = 3.00$, the area is approximately 0.9987.

The percentage between these Z-scores is: $0.9987 - 0.0013 = 0.9974$ So, approximately 99.74% of the scores fall between 188 and 872.

The answer is 99.7% (rounded to one decimal place).

(b) What percentage of standardized test scores is less than 188 or greater than 872?

This is the complement of the answer in part (a), as the percentage of scores outside the range 188 to 872 is the complement of those inside it.

$100\% - 99.7\% = 0.3\%$

So, 0.3% of the scores are less than 188 or greater than 872.

(c) What percentage of standardized test scores is greater than 758?

For this, we need to find the percentage of scores above 758.

Step 1: Convert 758 to a Z-score:

$Z = \frac{758 - 530}{114} \approx 2.00$

Step 2: Look up the Z-score in the standard normal distribution table.

For $Z = 2.00$, the area to the left of the Z-score is approximately 0.9772. Therefore, the area to the right (which represents the scores greater than 758) is: $1 - 0.9772 = 0.0228$ Thus, approximately 2.3% of the scores are greater than 758.

Would you like more detailed steps or clarifications? Here are five related questions that can expand on this information:

1. How do we use Z-scores to compare different datasets?
2. What is the empirical rule and how does it relate to normal distributions?
3. How do you calculate probabilities for non-standard normal distributions?
4. Why are normal distributions so common in statistics?
5. How can you estimate probabilities without a Z-table?

Tip: When converting raw scores to Z-scores, always remember the standard deviation and mean are key for normalization!