In this 3-part question, you will convert raw ACT scores to Z-scores and answer some questions about them.

Convertact_scoresto Z-scores. Recall fromData Visualization(the second course in this series) that to standardize values (convert values into Z-scores, that is, values distributed with a mean of 0 and standard deviation of 1), you must subtract the mean and then divide by the standard deviation. Use the mean and standard deviation ofact_scores, not the original values used to generate random test scores.

Question 3a

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What is the probability of a Z-score greater than 2 (2 standard deviations above the mean)?

Question 3b

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What ACT score value corresponds to 2 standard deviations above the mean (Z = 2)?

Question 3c

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A Z-score of 2 corresponds roughly to the 97.5th percentile.

Useqnorm()to determine the 97.5th percentile of normally distributed data with the mean and standard deviation observed inact_scores.

What is the 97.5th percentile ofact_scores?

In this 4-part question, you willwrite the function to create the CDF for ACT scores.

Write the function that takes a value and produces the probability of an ACT score less than or equal to that value (the CDF). Apply this function to the range 1 to 36.

Question 4a

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What is the minimum integer score such that the probability of that score or lower is at least .95?

Your answer should be an integer 1-36.

Question 4b

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Useqnorm()to determine the expected 95th percentile, the value for which the probability of receiving that score or lower is 0.95, given a mean score of 20.9 and standard deviation of 5.7.

What is the expected 95th percentile of ACT scores?

Question 4c

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As discussed in the Data Visualization course, we can usequantile()to determine sample quantiles from the data.

Make a vector containing the quantiles forp <- seq(0.01, 0.99, 0.01), the 1st through 99th percentiles of theact_scoresdata. Save these assample_quantiles.

In what percentile is a score of 26?

Your answer should be an integer (i.e. 60), not a percent or fraction. Note that a score between the 98th and 99th percentile should be considered the 98th percentile, for example, and that quantile numbers are used as names for the vectorsample_quantiles.

Question 4d

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Make a corresponding set of theoretical quantiles usingqnorm()over the intervalp <- seq(0.01, 0.99, 0.01)with mean 20.9 and standard deviation 5.7. Save these astheoretical_quantiles. Make a QQ-plot graphingsample_quantileson the y-axis versustheoretical_quantileson the x-axis.

Which of the following graphs is correct?