Problem 20 - Algebra 2 and Trigonometry

Question

The table shows the number of correct answers on a test consisting of 15 questions. The table represents correct answers for a sample of the students who took the test. Find the standard deviation based on this sample. \(\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Correct } & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} & {14} & {15} \\ \hline \text { Frequency } & {2} & {1} & {3} & {3} & {5} & {8} & {8} & {5} & {4} & {1} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

The standard deviation of the sample is approximately 2.174.

Step 1: Calculate the Mean

To find the standard deviation, we first need to compute the mean (average) number of correct answers. Multiply each number of correct answers by its frequency and sum up these products: \[(6 \times 2) + (7 \times 1) + (8 \times 3) + (9 \times 3) + (10 \times 5) + (11 \times 8) + (12 \times 8) + (13 \times 5) + (14 \times 4) + (15 \times 1) = 301\] Next, divide by the total number of students, which is the sum of the frequencies \(2 + 1 + 3 + 3 + 5 + 8 + 8 + 5 + 4 + 1 = 40\). Thus, the mean is \( \frac{301}{40} = 7.525 \).

Step 2: Find Variance Components

Calculate the square of the difference between each data point and the mean. Multiply each squared difference by its frequency. Sum these products to get the numerator for variance:\[\begin{align*}&(6-7.525)^2 \times 2 + (7-7.525)^2 \times 1 + (8-7.525)^2 \times 3 + (9-7.525)^2 \times 3+ \&(10-7.525)^2 \times 5 + (11-7.525)^2 \times 8 + (12-7.525)^2 \times 8 + (13-7.525)^2 \times 5+ \&(14-7.525)^2 \times 4 + (15-7.525)^2 \times 1 = 184.475 \end{align*}\]

Step 3: Calculate The Variance

To find the sample variance, divide the sum of the squared differences by the number of data points minus one: \[\frac{184.475}{39} \approx 4.729 \]

Step 4: Compute Standard Deviation

Take the square root of the variance to find the standard deviation: \[\sqrt{4.729} \approx 2.174 \]

Step 5: Conclusion

The standard deviation of the number of correct answers is approximately 2.174.

Key Concepts

Mean Calculation

Finding the mean is your starting point when calculating the standard deviation. The mean, or average, gives us a sense of the typical value within a data set. In this exercise, each number of correct answers on a test is paired with a frequency, which is how often that score appeared among the students. To calculate the mean:

• Multiply each score by its frequency. For example, the score of 6 appeared 2 times, so \(6 \times 2 = 12\).
• Continue this step for each score and frequency pair: \(7 \times 1, 8 \times 3,\) and so on.
• Sum all these products to find the total: 301 in this case.
• Finally, divide this sum by the total number of occurrences (sum of frequencies), which is the total number of students. Here, that number is 40.

Thus, the mean number of correct answers is \( \frac{301}{40} = 7.525\). Calculating this mean provides a basis for further statistical analysis, such as variance and standard deviation.

Variance

Variance measures how much the scores in a data set differ from the mean. It's an essential step to understanding data variability. After finding the mean, you need to determine how each score deviates from it. To do so, follow these steps:

• Subtract the mean from each score, which gives you the difference for each score. For example, for a score of 6, the difference is \(6 - 7.525 = -1.525\).
• Square each difference to ensure all values are positive, which helps avoid cancelation when summing.
• Multiply each squared difference by the frequency of that score, which adjusts for how common each score is.
• Sum all these products to find the total variability, which in this exercise is 184.475.

Variance itself is found by dividing this total by one less than the number of data points: \(\frac{184.475}{39} \approx 4.729\). Variance provides a picture of spread and variability about the mean, crucial for determining consistency in scores.

Frequency Distribution

Understanding frequency distribution in this context helps visualize not just average performance, but also the spread of scores. Frequency distribution refers to a table that lists each data point or outcome, with the number of times it occurs in the dataset. In this exercise:

• Each score between 6 and 15 represents how students performed on the test.
• The frequency, paired with each score, tells us how many students achieved that exact score.
• This table format makes it easy to see which scores were most and least common.
• It also helps in quantitatively assessing how data is distributed across possible outcomes.

Frequency distributions are foundational in statistics for summarizing large datasets and setting the stage for further analysis, like finding the mean or variance. It's akin to a snapshot of data behavior within any given dataset.

Algebra

Algebra plays a significant role in calculating statistics such as mean or standard deviation. It involves mathematical operations critically needed to find these values. For example:

• Using algebraic expressions, the mean is calculated as the sum of all values divided by the total count.
• The variance is derived using the formula, which involves squaring differences and working through multiplications of those differences by their frequencies.
• Other algebraic manipulations such as factoring, summing sequences, or simplifying complex expressions are also applied, making it a valuable tool in dealing with various statistical applications.

Recognizing algebra's power lets you solve structured problems in statistical methods. Its flexibility allows us to manipulate numbers and formulas to find precise solutions when analyzing data.