Statistics：練習問題

Which of the following events is an impossible event when rolling a standard fair six-sided die once?

The mean weight of a group of \(10\) boys is \(55\text{ kg}\), and the mean weight of a group of \(20\) girls is \(46\text{ kg}\). What is the mean weight of the whole group of \(30\) students?

A set of 40 students' scores is initially organized into a frequency distribution table with the following class intervals and frequencies:

Using the class marks for estimation, the mean score was estimated.

However, when the actual raw data was later obtained, it was discovered that the specific scores were highly skewed within their intervals:

• All 10 scores in the first class (\(10 \le x < 20\)) were exactly 12.
• All 15 scores in the second class (\(20 \le x < 30\)) were exactly 24.
• All 15 scores in the third class (\(30 \le x < 40\)) were exactly 34.

Calculate the actual mean score of the 40 students.

The heights of \(30\) students in a class are recorded and organized into the frequency distribution summarized below:

Height (\(h\) cm): \(140 \le h < 150\), \(150 \le h < 160\), \(160 \le h < 170\), \(170 \le h < 180\)
Frequency: \(6\), \(12\), \(9\), \(3\)

Identify the modal class of this data set.

A group of \(100\) students measured their heights (to the nearest cm). The data was organized into the following frequency distribution table:

Height (cm) | Frequency
\(140 \le h < 150\) | \(12\)
\(150 \le h < 160\) | \(45\)
\(160 \le h < 170\) | \(25\)
\(170 \le h < 180\) | \(18\)

Identify the median class and the class containing the upper quartile (\(Q_3\)).

Calculate the class mark of the class interval \(150 \le x < 160\).

In a histogram with equal class widths, a bar representing a frequency of \(24\) has a height of \(6\text{ cm}\). If another bar in the same histogram represents a frequency of \(36\), calculate the height of this bar in \(\text{cm}\).

The duration of 40 phone calls is distributed in groups: 5 calls last \(0 < t \le 2\) minutes, 15 calls last \(2 < t \le 4\) minutes, and 20 calls last \(4 < t \le 6\) minutes. Estimate the mean duration of a phone call, rounded to two decimal places.

The prices (in dollars) of minor repair parts for a bicycle at a local shop are: 8, 12, 10, 15, 12, 9, 11.

(a) Calculate the mean price of these parts.
(b) Determine the median and the mode of the prices.
(c) If the price of each part is increased by 2 dollars due to inflation, what will be the new mean price?

A data set \(X\) consists of five distinct positive integers. The median of the set is 10, the mean is 12, and the unique mode is 8.

(a) Determine the five integers in the data set \(X\).

(b) A new data set \(Y\) is created using the linear transformation \(y = 2x - k\). If the mean of the new set \(Y\) is \(20\), find the value of the constant \(k\).

(c) Calculate the mode of the new data set \(Y\).