Final answer:
To create a graph for the normal distribution, draw a bell-shaped curve with a mean of 83 and standard deviation of 5. The area to the right of a specific value represents the probability of scores higher than that value. The z-score is calculated to determine how many standard deviations a score is from the mean.
Step-by-step explanation:
Understanding the Normal Distribution
When discussing exam scores that are normally distributed, we imagine a symmetrical bell-shaped curve centered around the mean, with the standard deviation indicating how spread out the scores are around the mean. The total area under the curve represents the probability of all possible outcomes and equals 1.
Sketching the Normal Curve
To sketch a normal distribution with a mean (μ) of 83 and a standard deviation (σ) of 5, draw a symmetrical bell-shaped curve on the graph. Mark the horizontal axis with the mean at the center (83) and additional marks at intervals of the standard deviation (78, 73, etc., to the left and 88, 93, etc., to the right).
Shading Areas Above a Value
If we want to show the area above a certain value, for example, an exam score of 90, shade the region under the curve to the right of the 90 mark. This shaded area represents the probability of a student scoring above 90 in the class. The larger the area, the higher the probability.
Calculating a Z-Score
To calculate the z-score, use the formula: z = (X - μ) / σ. A z-score indicates how many standard deviations an individual score is from the mean. For example, if a student scored 90, their z-score would be (90 - 83) / 5 = 1.4.
Interpreting the Z-Score
A z-score of 1.4 means that the student's score is 1.4 standard deviations above the class mean, which is a high score compared to the rest of the class.