# Mathematics homework: normal probability curve

INSTRUCTION

In this unit, you will investigate the normal probability curve (the bell curve). Many variables, such as height and weight are “normally distributed.” This means, for example, that if you were to collect 10,000 female adult human heights, the histogram of that data would be shaped like a “bell” (with “most” of the data near the center or mean).

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Use the following z table portion to assist you with answering the Discussion topics. There is a full z table in Doc Sharing.

Different university departments use different tests to compare student performance and to determine graduate admission status. Three such tests are the GMAT, the LSAT, and the GRE.

- Across the USA, results for these exams are normally distributed. What does that mean and why is this the case?
- If you were to create a histogram of all GRE scores, what would you expect the histogram to look like? Would it be symmetrical? Would it be bell shaped? How many modes would it likely have? Would it be skewed?
- Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the 68-95-97.7 (Empirical) Rule to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500? What percentage of students will get a score below 275? Is a score below 275 significantly different from the mean? Why or why not?
- Choose any GRE score between 200 and 800. Be sure that you do not choose a score that a fellow student has already selected. Using your chosen score, how many standard deviations from the mean is your score? (This value is called the z-value). Using the table above (or the z table in Doc Sharing), what percentage of students will likely get a score below this value? What percentage of students is likely to get a score above this value?
- Review the student post and evaluate their solutions for using the 68-95-97.7 (Empirical) Rule to determine the percentage of GRE scores between 350 and 650. Are the student’s calculations correct? If yes, note this and if not correct them with an example. Next, explain to the student why 50% of the scores are above 500 and why 50% are below (approximately).
- Review the student’s GRE score choice from number 4 above. Are the student calculations correct? Include the student’s calculations in your response and note any issues if discovered. Then, offer the student a second example using any other value between 300 and 500. Be sure to explain all the steps in your example to the student and to show all work.

Choose any two classmates and review their main posts.

- Review the student post and evaluate their solutions for using the 68-95-97.7 (Empirical) Rule to determine the percentage of GRE scores between 350 and 650. Are the student’s calculations correct? If yes, note this and if not correct them with an example. Next, explain to the student why 50% of the scores are above 500 and why 50% are below (approximately).
- Review the student’s GRE score choice from number 4 above. Are the student calculations correct? Include the student’s calculations in your response and note any issues if discovered. Then, offer the student a second example using any other value between 300 and 500. Be sure to explain all the steps in your example to the student and to show all work.

STUDENTS LOUISE POST

- The results are normally distributed because the data is centered around the mean so this data is equally distributed to the left and right of the mean and creates a mirror image.
- The histogram should have a bell curve this would happen due to the data being normally distributed. It would not be skewed due to the most of the data being centered. There would be one mode which would be the peak top.
- 95% of the students will score between 350-650, 50% of the students will score above 500. A score below 275 is not very likely and 275 is very different from the mean it is 225 points below the mean.
- The GRE score of 425 is 33% likely and a score below 425 is 77% likely. Z score would be -0.3.

MATTHEW POST

Normally distributed data is data where most of the data is centered around the mean, as you get further from the mean the chances of it occurring are less likely.

Since the data is normally distributed, it will be symmetrical with both sides being a mirror image of each other. It would have a bell shape due to the normal distribution of data focused around the mean. There will be one mode which is the peak of the curve. The data will not be skewed in a normal distribution chart, the center is focused around the mean and both sides are symmetrical. In a normally distributed graph the mean, median and mode are the same.

68% of the data will fall within one standard deviation of the mean, 95% will fall within two standard deviations of the mean, 99.7% will fall within three standard deviations of the mean.

68% of the score is between 500 – 75= 425 and 500 + 75= 575

95% of the score will fall between 500- 2(75) = 350 and 500+ 2(75) = 650

99.7% of the score will fall between 500- 3(75) = 275 and 500+3(75) = 725

Due to the normal distribution of data 50% will be above 500 and 50 % below 500. A score below 275 is not close to the mean and out of the normal data range so it is significantly different. A score below 275 is not likely because 99.7% of students are between 275 and 725.

Z=300-500/75= -2.66 standard deviation below the mean.

0.0039=0.39% will have a score below this, while 99.64% will have a score above.

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