數值量度、圖表與圖示：练习题

A data set of annual salaries (in \(\$\)) has a strong positive skew.
Which measure of central tendency and which measure of spread would be most appropriate to summarize this data?

In a box and whisker plot, the lower quartile is 24, the median is 35, and the upper quartile is 42.
An outlier is defined as any value more than \(1.5 \times \text{IQR}\) above the upper quartile or below the lower quartile.
Determine the boundaries used to identify outliers for this data set.

A dataset of test scores for \(20\) students has a mean of \(65\) and a standard deviation of \(10\). During a review, it is discovered that one score was incorrectly recorded as \(40\) instead of \(80\). Calculate the corrected standard deviation of the dataset to two decimal places.

Which of the following features of a histogram is essential to ensure it correctly represents data with unequal class intervals?

A set of data regarding the weights of 15 samples has a median of 42 g and an interquartile range (IQR) of 12 g. The lower quartile \( Q_1 \) is 35 g. An outlier is defined as any value more than \( 1.5 \times \text{IQR} \) outside the quartiles.

If every value in the dataset is increased by 5 g and then multiplied by 1.1, what are the new boundaries for identifying outliers?

A data set consists of the following values: 12, 15, 15, 17, 18, 20, 22. Identify the median and the mode of this data set.

The weights of 100 apples are summarized in a histogram with unequal class intervals. The frequency of apples weighing between 120g and 150g is 45. Calculate the frequency density for this class interval.

A student calculates the upper outlier boundary for a data set using the formula \(Q_3 + 1.5 \times (Q_3 - Q_1)\). For a distribution where \(Q_1 = 42\), the median is 50, and the data is heavily positively skewed, explain why an extreme value might be identified as an outlier even if it is a valid member of the population.

A set of raw data representing the daily rainfall (in mm) at a weather station over 10 days is as follows:
4.2, 5.1, 0.0, 12.8, 3.5, 6.4, 4.8, 20.5, 5.5, 3.2

(a) Calculate the mean and the standard deviation for this dataset.
(b) An outlier is defined as any value that is more than 1.5 times the Interquartile Range (IQR) above the upper quartile or below the lower quartile. Using this definition, determine if the value 20.5 is an outlier.

A researcher is studying the waiting times, in minutes, for patients at two different medical clinics, Clinic A and Clinic B. The data for Clinic A is summarized in the following table:

Waiting Time (t, minutes) | Frequency
\(0 \le t < 10\) | 12
\(10 \le t < 20\) | 28
\(20 \le t < 30\) | 35
\(30 \le t < 50\) | 15
\(50 \le t < 80\) | 10

(a) Calculate the frequency density for the class interval \(30 \le t < 50\) and the class interval \(50 \le t < 80\). (2 points)
(b) Estimate the median waiting time for Clinic A using linear interpolation. (3 points)
(c) For Clinic B, the median waiting time is 22 minutes and the interquartile range is 12 minutes. Compare the waiting times of Clinic A and Clinic B using your answer to part (b) and the fact that the interquartile range for Clinic A is approximately 18.5 minutes. (2 points)