Describing distributions of one quantitative variable：練習問題

In a symmetric, bell-shaped distribution, approximately what percentage of the data falls within two standard deviations of the mean, according to the Empirical Rule?

Two students from different schools want to compare their performance on a math placement exam.
Student A scored 82 on a test with a mean of 70 and a standard deviation of 8.
Student B scored 78 on a test with a mean of 66 and a standard deviation of 5.
Comparing their standardized scores (z-scores), which student performed better relative to their respective school?

A researcher collects data on the following variables from a group of college students. Which of these variables is considered categorical?

A teacher calculates the summary statistics for a test and finds the first quartile (\( Q_1 \)) is 65 and the third quartile (\( Q_3 \)) is 85. Using the 1.5 \( \times \) IQR rule, which of the following scores would be identified as an outlier?

Consider a data set where every observation is exactly the same value, 10. What is the standard deviation of this data set?

A dataset of test scores has a mean of \( 75 \) and a standard deviation of \( 10 \). If a student earns a score of \( 85 \), calculate the student's z-score and explain its meaning in terms of standard deviations from the mean.

A small company records the number of items returned by customers each day for 10 days. The data collected are:
\( 1, 2, 2, 3, 4, 4, 5, 6, 7, 15 \).

(a) Find the median of the data set.
(b) Find the first quartile \( Q_1 \), the third quartile \( Q_3 \), and calculate the interquartile range (IQR).
(c) Determine if the value \( 15 \) is an outlier using the \( 1.5 \times \text{IQR} \) rule. Show your calculations.