Mat 151m6a114 Points Made On His Accounting Midterm

Mat 151m6a114 Points1bob Made 116 On His Accounting Midterm And 82 On

Mat 151m6a114 points 1.bob made 116 on his accounting midterm and 82 on his biology midterm. For the accounting midterm, the mean score was 102 with standard deviation 14. For the biology midterm, the mean score was 70 with standard deviation of 5. Which score was higher with respect to the rest of the class? Explain your answer.

Answer questions 2 - 5 dealing with the life expectancy of light bulbs whose lifetimes are normally distributed with a mean life of 750 hours and with a standard deviation of 80 hours. 2. What percent of light bulbs will last longer than 870 hours? 3. What percent of light bulbs will last between 730 hours and 850 hours? 4. What percent of light bulbs will last less than 770 hours? 5. If the quality control program of the company can consistently eliminate the worst 10% of the bulbs manufactured, the manufacturer can safely offer customers a money-back guarantee on all lights that fail before __________ hours of burning time.

An investigator polls a representative sample of common cold sufferers, asking them to estimate the number of hours of physical discomfort caused by their most recent cold. Their estimates approximate a normal curve with a mean of 83 hours and a standard deviation of 20 hours. 6. What proportion of sufferers estimate that their colds lasted for longer than forty-eight hours? 7. What proportion suffered for fewer than 61 hours? 8. What proportion suffered for between one and three days? 9. What is the estimated number of hours for the shortest-suffering 10 percent?

A medical researcher wishes to concentrate on the 20 percent who suffered the most. She will work only with those who estimate that they suffered for more than _____ hours.

A group of statistics students took a midterm exam. The class mean was 72 and the standard deviation was 3.2. The instructor decided to "curve" the grades. Assuming that the distribution of grades was normal, the instructor decided to assign grades the following way: Students whose test grades are within 1/2 standard deviation of the class mean will receive C's. B's and D's will be given to scores from (and including) 1/2 standard deviation up to (and including) 1 1/2 standard deviations above and below the mean respectively. A's and F's will be given to scores above and below 1 1/2 standard deviations of the mean respectively. 10. What percentage of students will receive each grade? 11. What grade would a student with a score of 70 make? 12. What grade would a student with a score of 86 make? 13. Devise a grading scale for the test. Values should be rounded to the nearest ones' place. Remember that no grades may exist in more than one category.

Paper For Above instruction

The analysis of scoring in various academic and practical contexts relies heavily on understanding standardized scores, or z-scores, which express individual performances relative to their respective distributions. This essay explores the comparison of student scores, the life expectancy of light bulbs, discomfort duration in cold sufferers, and grading scales, examining how z-scores facilitate interpretation and decision-making in these scenarios.

Comparing Student Performance

Bob's scores in accounting and biology are evaluated relative to their class distributions. His accounting score of 116 has a mean of 102 and a standard deviation of 14, yielding a z-score of (116 - 102)/14 ≈ 1.00. His biology score of 82, with a mean of 70 and a standard deviation of 5, results in a z-score of (82 - 70)/5 = 2.40. The higher z-score indicates Bob's biology performance was more exceptional relative to his peers. Despite a higher raw score in accounting, his relative standing is better in biology, emphasizing the importance of standardized scores in performance evaluation.

Life Expectancy of Light Bulbs

The normal distribution assumption allows calculation of probabilities for bulb lifetimes. For bulbs lasting longer than 870 hours, the z-score is (870 - 750)/80 = 1.625. Consulting standard normal tables, approximately 5.2% of bulbs exceed this lifespan. Similarly, for the range between 730 and 850 hours, the z-scores are -0.875 and 1.375, corresponding to probabilities of about 19.0% and 91.0%. Thus, approximately 72% of bulbs fall within this range. For bulbs lasting less than 770 hours, the z-score is (770 - 750)/80 = 0.25, with about 40.1% of bulbs shorter than this time. If the manufacturer eliminates the lowest 10%, they can guarantee bulbs that fail before the 750 - 1.28*80 ≈ 646 hours, aligning with the lower 10th percentile.

Duration of Cold Sufferers' Discomfort

The cold discomfort duration follows a normal distribution with a mean of 83 hours and standard deviation of 20 hours. The proportion experiencing more than 48 hours is calculated with z = (48 - 83)/20 ≈ -1.75, corresponding to about 4.0% exceeding 48 hours. Fewer than 61 hours corresponds to z = (61 - 83)/20 ≈ -1.10, with approximately 13.6% reporting shorter durations. Between one and three days (24 to 72 hours) involve z-scores of approximately -2.35 and -0.55, equating to about 1.0% and 28.2%. The shortest 10% of sufferers experienced durations below the 10th percentile, corresponding to a z-score of about -1.28, approximately 63 hours.

The medical researcher aims to focus on the top 20% of sufferers, which starts at the 80th percentile, with a z-score of approximately 0.84. Applying this to the distribution, the cutoff duration is 83 + 0.84*20 ≈ 100.8 hours.

Grading Students Based on Normal Distribution

The students' scores distribute normally with a mean of 72 and a standard deviation of 3.2. The grading scheme categorizes scores within specific z-value ranges. Scores within ±0.5 standard deviations (69.2 to 74.8) correspond to approximately 38.3% of students receiving C grades. Scores between 0.5 and 1.5 standard deviations above or below the mean cover about 43.2%, split into B's and D's. Scores above +1.5 standard deviations (greater than 76.4) will receive A's, accounting for about 6.7%, and those below -1.5 standard deviations (less than 68.8) will receive F's, also approximately 6.7%. A student scoring 70 is within 0.5 SD below the mean and would thus receive a C, while a score of 86 exceeds +1.5 SD above the mean, earning an A. Based on these, a comprehensive grading scale can be devised to ensure clarity and non-overlapping categories.

Conclusion

The use of standardized scores provides critical insights into performance and expectations across various domains. From academic assessments to quality control and health data, z-scores enable fair comparisons and informed decisions, illustrating their foundational role in statistical analysis.

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