Standard Deviation and Conditional Probability (Extended)：練習問題

The box-and-whisker plot shown represents the distribution of test scores for a class of students. Based on the diagram, determine the interquartile range (IQR) of the scores.

A set of data has a mean of \( 50 \) and a standard deviation of \( 8 \). If every value in the data set is multiplied by \( 1.5 \) and then increased by \( 10 \), calculate the new mean and the new standard deviation.

For a set of \( 50 \) observations, it is given that \( \sum x = 400 \) and \( \sum x^2 = 3450 \). Calculate the standard deviation of these observations, correct to two decimal places.

Class X has a mean score of \( 60 \) with a standard deviation of \( 10 \). Class Y has a mean score of \( 60 \) with a standard deviation of \( 15 \). If Student A from Class X and Student B from Class Y both score \( 75 \), determine who performed better relative to their own class by calculating their z-scores.

A set of data consists of the following values: 5, 8, 12, 15, 15, 18, 22, 25.
(a) Determine the median of the data set.
(b) Calculate the interquartile range (IQR) for this data set.

John received his results for two subjects. The class parameters and John's scores are given below:
- Mathematics: John scored \( 78 \); Class mean \( = 70 \), Standard deviation \( = 5 \).
- Science: John scored \( 80 \); Class mean \( = 72 \), Standard deviation \( = 8 \).
(a) Calculate the standard scores (\( z \)-scores) for Mathematics and Science. In which subject did John perform better relative to his class?
(b) In History, the class mean is \( 65 \) and the standard deviation is \( 10 \). If John wants to achieve the same relative performance in History as he did in Mathematics, what raw score must he obtain?