Read Chapter 3 In Cognition

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Read Chapter 3 in Cognition: PSY 3200 Unit #3 AS: Calculating Percentages You will be given a series of questions regarding a normal distribution, where you need to determine the percentage above or below specific raw scores or compute the raw scores corresponding to particular percentages. For each question, provide the detailed calculation process, including the Z score, the shaded area on the curve, the precise percentage from the normal curve table, and the final answer. Submit your responses typed in the assigned Word document via Blackboard. The context involves a group of students taking a statistics exam with an average score (M) of 90 and a standard deviation (SD) of 7.8. Use your normal curve table to answer the specific questions below, identifying raw scores, Z scores, the shaded area on the curve, and exact percentages. Each question is worth 5 points.

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The task requires analyzing a normal distribution of exam scores and answering 15 questions related to percentages and raw scores within this distribution. The questions involve calculating the percentage of students who scored above or below certain scores, determining raw scores needed to fall within specific percentiles, and identifying score ranges that represent particular proportions of the population. These calculations demand converting raw scores to Z scores, visualizing the normal curve and shading relevant areas, and consulting the normal curve table to find precise percentages. The process also includes reversing Z score calculations to find raw scores from given percentages or percentiles.

Understanding the application of Z scores is fundamental: they measure how many standard deviations a raw score is from the mean. For example, if a student scores 99 on the exam, we convert this raw score to a Z score using the formula Z = (X - M)/SD, which yields the Z score (+1.15 in this case), indicating the position relative to the mean. By referencing the normal curve table for this Z score, the exact percentage of students scoring above this score can be determined. Conversely, if a specific percentage is given (e.g., the top 8%), the corresponding Z score can be found from the table, which can then be converted into a raw score.

When calculating percentages, visualizing the normal curve and shading the relevant area enhances comprehension. For example, to find the percentage of scores between two raw scores, convert both to Z scores, look up their corresponding areas in the table, and compute the difference. For instance, the middle 40% of scores can be located by identifying Z scores that mark the upper and lower bounds of that interval, then converting these Z scores back into raw scores. This process emphasizes the symmetrical nature of the normal distribution, where percentages between positive and negative Z scores mirror each other.

Similarly, for percentile calculations, converting a percentile into a Z score involves locating the cumulative area (e.g., 90th percentile corresponds to a Z score of approximately +1.28). Using this Z score, the raw score can be calculated. For example, to reach the 90th percentile with an exam mean of 90 and SD of 7.8, Z = 1.28; the raw score (X) is then computed as X = (Z * SD) + M. Inversely, raw scores can be converted into percentile ranks using Z scores and lookups from the normal curve table.

The calculations also include determining the proportion of the population within specific score ranges—like scores between 82 and 107—by converting both scores into Z scores, obtaining their respective cumulative percentages, and subtracting to find the proportion in that interval. For scores below a certain raw score, the Z score is negative; for scores above, it is positive. Careful attention is required to interpret the exact shaded areas, especially when dealing with scores in the tail ends of the distribution.

Finally, probability concepts are intertwined with these calculations: the likelihood of a randomly selected student scoring above or below a given raw score can be expressed as a probability, derived by the ratio of the relevant area under the normal curve. For instance, the probability of scoring more than 96, with the mean and SD given, can be calculated through Z scores and referencing the normal curve table. The understanding of probability lays the groundwork for inferential statistics procedures, such as hypothesis testing and confidence intervals.

References

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