According to ACT, results from the 2008 ACT testing found...

According to ACT, results from the 2008 ACT testing found that students had a mean reading score of 21.4 with a standard deviation of $6.0 .$ Assuming that the scores are normally distributed, a. find the probability that a randomly selected student had a reading ACT score less than $20 .$ b. find the probability that a randomly selected student had a reading ACT score between 18 and 24 c. find the probability that a randomly selected student had a reading ACT score greater than 30 d. find the value of the 75th percentile for ACT scores.

According to ACT, results from the 2008 ACT testing found that students had a mean reading score of 21.4 with a standard deviation of $6.0 .$ Assuming that the scores are normally distributed,
a. find the probability that a randomly selected student had a reading ACT score less than $20 .$
b. find the probability that a randomly selected student had a reading ACT score between 18 and 24
c. find the probability that a randomly selected student had a reading ACT score greater than 30
d. find the value of the 75th percentile for ACT scores.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, forming a bell-shaped curve. It is defined by its mean and standard deviation, and most values cluster around the mean, with probabilities tapering off symmetrically on both sides. Many statistical methods assume normality due to its well-known properties and behavior.

Z-Scores

A z-score indicates how many standard deviations an observation is from the mean of the distribution. By converting raw scores to z-scores, one can utilize the standard normal distribution to compute probabilities, compare scores from different normal distributions, and standardize data for analysis.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a distribution represents the probability that a random variable takes on a value less than or equal to a specific value. In the context of the normal distribution, the CDF is used to find cumulative probabilities, which are essential for computing probabilities for ranges of values and tail probabilities.

Percentiles

Percentiles are measures that indicate the value below which a given percentage of observations in a dataset fall. In a normal distribution, percentiles can be calculated by finding the corresponding z-score from the inverse CDF, which then helps to determine the raw score associated with a specific percentile.

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