Problem 65 - Calculus and its applications

Question

Test score distribution. In a large class, student test scores had a mean of \(\mu=76\) and a standard deviation \(\sigma=7\) a) The top \(12 \%\) of students got an \(A\). Find the minimum score needed to get an \(\mathrm{A}\) (round to the appropriate integer). b) The top \(75 \%\) of students passed. Find the minimum score needed to pass (round to the appropriate integer).

Step-by-Step Solution

Answer

a) 84 b) 71

Step 1: Define the Problem for Part (a)

We need to find the minimum score needed to get an A, given the top 12% of students received an A. This means we are looking at the cutoff score for the 88th percentile in a normal distribution with mean \(\mu=76\) and standard deviation \(\sigma=7\).

Step 2: Find the Z-Score for the 88th Percentile

Using the Z-table or a standard normal distribution calculator, we find the Z-score that corresponds to the cumulative probability of 0.88. The Z-score for 0.88 is approximately 1.175.

Step 3: Compute the Minimum Score Using the Z-Score

We use the Z-score formula to find the minimum score: \[ x = \mu + Z\cdot\sigma = 76 + 1.175\cdot7 \approx 84.225 \]Rounding this to the nearest whole number, the minimum score required to get an A is 84.

Step 4: Define the Problem for Part (b)

For part b, we need to determine the minimum score needed to pass, which corresponds to the 25th percentile (since we want the top 75% of students).

Step 5: Find the Z-Score for the 25th Percentile

Using the Z-table or a standard normal distribution calculator, we find the Z-score for a cumulative probability of 0.25. The Z-score for 0.25 is approximately -0.675.

Step 6: Compute the Minimum Passing Score Using the Z-Score

Apply the Z-score formula to find the minimum score that constitutes the 25th percentile:\[ x = \mu + Z\cdot\sigma = 76 + (-0.675)\cdot7 \approx 71.275 \]Rounding this to the nearest whole number, the minimum passing score is 71.

Key Concepts

Percentiles

Understanding percentiles is crucial when analyzing data, especially in a normal distribution. Percentiles indicate the relative standing of a value within a data set.
*They tell us how a particular score compares to the rest of the data.*
For instance, if a test score is in the 88th percentile, it means that the score is higher than 88% of the other scores in the distribution.

• The **88th percentile** implies that 12% of scores are above and 88% are below.
• The **25th percentile**, also known as the first quartile, means 25% of scores are below and 75% are above.

In the context of the exercise, calculating percentiles helped identify the minimum scores needed for different academic achievements such as getting an 'A' or merely passing. Percentiles provide a simple way to interpret individual scores in the context of the entire class performance.

Z-score calculation

Calculating the Z-score is essential for finding where a specific score stands within the normal distribution. The Z-score represents the number of standard deviations a particular data point is from the mean.
*It's a tool used to convert test scores into a standard form that reveals comparative standings.*

• For finding the **88th percentile**, we determine a Z-score by setting the cumulative probability at 0.88.
• Conversely, for the **25th percentile**, a Z-score corresponds to a cumulative probability of 0.25.

Once you have these Z-scores, the test scores can then be calculated using the formula:\[ x =  \mu + Z \cdot \sigma \]Here, \(\mu\) is the mean of the distribution, \(\sigma\) is the standard deviation, and \(Z\) is the Z-score. This formula helps translate Z-scores back into real-world scores and is the final step in determining thresholds for grades and passing.

Test score analysis

With test score analysis, the goal is to understand the distribution of scores and make decisions based on statistical measurements. *This involves applying percentiles and Z-scores to draw meaningful conclusions.*

• **Identifying top performers:** By analyzing the top 12% of scores, an educator can determine what scores qualify as an 'A'.
• **Evaluating pass/fail thresholds:** Calculating the cut-off for the 25th percentile allows educators to set a fair and consistent passing score.

Analysts can better understand class performance as a whole or individual performance through this process. This approach not only identifies high-achievers but also ensures that grading standards are applied justly across all students. It's a valuable practice that supports objective educational decisions, based on robust and replicable statistical methods.