In 2001, the national mean ACT Math score was 20.7, with an...

In 2001, the national mean ACT Math score was 20.7, with an approximate standard deviation of 6. ACT scores in the general population have a normal distribution. (a) Approximately what percentage of the 2001 scores were higher than 26.7? (b) Approximately what ACT Math score would one need to make in 2001 to be ranked among the top 2.5% of all who took the test? (c) Writing to Learn Mean ACT scores are published state by state. If we add up the 50 state means and divide by 50, will the result be a good estimate for the national mean score? Explain your answer.

In 2001, the national mean ACT Math score was 20.7, with an approximate standard deviation of 6. ACT scores in the general population have a normal distribution.
(a) Approximately what percentage of the 2001 scores were higher than 26.7?
(b) Approximately what ACT Math score would one need to make in 2001 to be ranked among the top 2.5% of all who took the test?
(c) Writing to Learn Mean ACT scores are published state by state. If we add up the 50 state means and divide by 50, will the result be a good estimate for the national mean score? Explain your answer.

Key Concepts

Percentiles and Quantiles

Percentiles refer to the values below which a given percentage of observations in a dataset falls. In the context of a normal distribution, quantiles (or inverse percentiles) are used to determine the value corresponding to a specific cumulative probability. This concept is important when determining thresholds for ranking, such as the score required to be in the top 2.5% of a population.

Z-scores

A z-score represents the number of standard deviations a data point is from the mean. It is used to standardize scores from different normal distributions, making it possible to calculate probabilities. By converting a raw score to a z-score, one can use standard normal distribution tables to determine the likelihood of a value occurring within the distribution, which is essential for answering questions about proportions and percentiles.

Weighted vs. Unweighted Means

When combining averages from different groups, it is important to know whether the groups are of equal size. A simple average of the means (an unweighted mean) will not accurately reflect the overall population mean if the groups (such as states) have different sample sizes. A weighted mean, which accounts for the varying sizes of each group, provides a more accurate estimate of the national average. This concept is critical for understanding potential biases when aggregating subgroup statistics.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by its mean and standard deviation and is widely used in statistics because of the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution, regardless of the original distributions. This makes it a fundamental tool for assessing probabilities and making inferences about population parameters.

Standard Deviation

Standard deviation measures the dispersion or spread of a set of data values relative to the mean. In the context of a normal distribution, it indicates how concentrated the data is around the mean. A smaller standard deviation means that the values are closely clustered around the mean, while a larger standard deviation means they are more spread out. This concept is crucial for standardizing values and calculating probabilities.

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