A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of  20  to  80 . In a recent year, the national mean score for the writing section was  48.4 , with a standard deviation of  9.1 . Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year.   (a) According to Chebyshev's theorem, at least ?56%75%84%89% of the scores lie between 30.2 and 66.6 . (b) According to Chebyshev's theorem, at least 8/9 (about 89%) of the scores lie between     and    . (Round your answer to 1 decimal place.) (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the scores lie between     and    . (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the scores lie between 30.2 and 66.6 .

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A: Solution

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.