Mean 81, Median 83, Mode 80, Standard Deviation 9, First Qua

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Analyze the given statistical data to answer the following questions: First, interpret the descriptive statistics provided, including measures of central tendency and dispersion, for the students' scores. Then, examine the specific percentile and quartile information to determine the distribution characteristics. Next, apply this understanding to solve the questions about student scores and distributions, including the calculation of specific percentiles, understanding percent exceeding certain values, and estimating total scores based on the number of students. Additionally, interpret the normal distribution regarding coffee prices, calculating the probability of sales within certain price ranges using the properties of the normal distribution.

Paper For Above instruction

The statistical data provided offers an extensive overview of students’ performance and coffee pricing in two distinct contexts. The first context involves understanding scores on a student assessment, summarized by measures such as the mean, median, mode, standard deviation, and quartiles. The second pertains to the distribution of coffee prices, modeled as a normal distribution, which allows for probability estimation within specific price ranges.

Analysis of Student Scores

The data indicates that the average (mean) student score is 81, with a median of 83, and a mode of 80, suggesting a slightly left-skewed distribution, where most students score around the low 80s but with some lower outliers. The standard deviation of 9 reflects the variability in students' scores, meaning most scores fall within nine points of the mean—roughly between 72 and 90. The quartile data specifies that 25% of students scored below 60, the median score was at 83, and 75% scored below 89. The 90th percentile at 95 indicates higher-performing students scoring well above the average.

Question 1 asks what score half of the students exceed. Since the median is 83, half of the students scored above 83, and half below. This means the answer is 83.

Question 2 inquires about the percentage of students who scored higher than 89. Given that 89 is the third quartile (Q3), approximately 25% of students scored above this mark because 75% scored below. Since the exact 90th percentile is at 95, slightly higher than 89, we estimate that about 10-12% of students scored above 89.

Question 3 considers students scoring higher than 95. Given the 90th percentile is at 95 and assuming a normal distribution, about 10% of the students scored above 95.

Question 4 involves calculating the total sum of scores for 100 students. With an average score of 81, total sum is 81 multiplied by 100, equaling 8,100.

Question 5 asks for the score representing one standard deviation above the mean. Calculating this: mean (81) + standard deviation (9) = 90. Therefore, a score of 90 represents one standard deviation above the mean.

Question 6 involves finding a score 1.5 standard deviations below the mean: 81 - (1.5 × 9) = 81 - 13.5 = 67.5. Rounded, this is approximately 68.

Analysis of Coffee Prices

The second distribution involves coffee prices, modeled as a normal distribution with a mean of $1.95 and a standard deviation of $0.23. Using the properties of normal distribution, we can answer probability-based questions.

Question 1 asks for the percentage of coffee sales between $1.75 and $2.15. First, calculate the z-scores:

• Z for $1.75 = (1.75 - 1.95) / 0.23 ≈ -0.87
• Z for $2.15 = (2.15 - 1.95) / 0.23 ≈ 0.87

Consulting standard normal distribution tables, the cumulative probability for Z = ±0.87 is approximately 0.8078. Therefore, the proportion of coffee sold between these prices is 2 × (0.8078 - 0.5) ≈ 2 × 0.3078 = 0.6156 or about 61.56%.

Question 2 asks for the percentage sold for more than $1.50, with Z = (1.50 - 1.95) / 0.23 ≈ -1.96. The cumulative probability for Z = -1.96 is roughly 0.025, so the remaining percentage (greater than $1.50) is 97.5%.

Question 3: for prices greater than $2.00, Z = (2.00 - 1.95)/0.23 ≈ 0.22. Looking up Z = 0.22 yields a cumulative probability of approximately 0.5871, meaning about 1 - 0.5871 = 0.4129 or 41.29% of the coffees are sold at prices exceeding $2.00.

Question 4 involves prices less than $2.20. Z = (2.20 - 1.95)/0.23 ≈ 1.09, with the cumulative probability at Z=1.09 being about 0.8621. Thus, approximately 86.21% of the cups are sold below $2.20.

Conclusion

The statistical insights from these two contexts illustrate how descriptive measures help interpret distribution characteristics. For the students' scores, median and quartiles reveal the distribution's skewness and spread, while standard deviation allows us to understand variability. For coffee prices, the normal distribution’s properties facilitate estimating the percentages of sales within specific price ranges, which is valuable for business analysis and planning. Both applications demonstrate the importance of statistical understanding in real-world decision-making.

References

• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2014). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Cengage Learning.
• Glantz, S. A. (2012). Primer of Biostatistics (7th ed.). McGraw-Hill.
• Weiss, N. A. (2005). Introductory Statistics (8th ed.). Pearson.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and Statistics for Engineers and Scientists (9th ed.). Pearson.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
• Chance, B. (2007). Operations Research: An Introduction (10th ed.). Thomson Brooks/Cole.
• Hogg, R. V., McKean, J., & Craig, A. T. (2013). Introduction to Mathematical Statistics (7th ed.). Pearson.