The length of songs in a collector's iTunes album collection is uniformly distributed from 2.2 to 3.1 minutes. Suppose we randomly pick seven albums from the collection. There are a total of 47 songs on the seven albums. a. In words, define the variable \( X \). A \( X= \) the length of a song, in minutes, in the collection. B \( X= \) the sum of the length of all songs, on ITunes. C \( X= \) the number of songs in the collection. D \( X= \) the number of songs on ITunes. b. Give the distribution of \( X \). \( X \sim \) \( \square \) U \( \sqrt{x} \) ( 2.2 3.1 (For numerical answers, enter values as integers or decimals.) c. Which of the following defines the variable \( X \) ? A \( X= \) the sum of the length, in minutes, of all the songs on itunes. B \( \bar{X}= \) the average length, in minutes, of all the songs on ITunes. c \( \bar{X}= \) the average length, in minutes, of the songs from a sample of seven albums from the collection.

Let's break down the problem step by step. ### Part a: Define the variable \( X \) We need to determine what \( X \) represents in the context of the problem. The problem states that the length of songs in a collector's iTunes album collection is uniformly distributed from 2.2 to 3.1 minutes. - **Option A** states that \( X \) is the length of a song, in minutes, in the collection. This is a reasonable definition since we are interested in the length of individual songs. - **Option B** states that \( X \) is the sum of the length of all songs on iTunes. This is not correct because \( X \) is defined as the length of a single song, not the total length. - **Option C** states that \( X \) is the number of songs in the collection. This is also incorrect as it does not pertain to the length of songs. - **Option D** states that \( X \) is the number of songs on iTunes. This is again not correct for the same reason as option C. Thus, the correct answer is: **A** \( X= \) the length of a song, in minutes, in the collection. ### Part b: Give the distribution of \( X \) The problem states that the length of songs is uniformly distributed between 2.2 and 3.1 minutes. Therefore, we can express the distribution of \( X \) as: \[ X \sim U(2.2, 3.1) \] So the answer is: \( X \sim U(2.2, 3.1) \) ### Part c: Which of the following defines the variable \( X \)? Now we need to analyze the options given: - **Option A** states that \( X \) is the sum of the length, in minutes, of all the songs on iTunes. This is incorrect because \( X \) is defined as the length of a single song, not the sum. - **Option B** states that \( \bar{X} \) is the average length, in minutes, of all the songs on iTunes. This is also incorrect because it does not define \( X \) but rather the average of all songs. - **Option C** states that \( \bar{X} \) is the average length, in minutes, of the songs from a sample of seven albums from the collection. This is also not a definition of \( X \) but rather a statistic derived from a sample. None of the options directly define \( X \) as it was defined in part a. However, if we are looking for the closest option that relates to the average length of songs, option C is the most relevant, but it still does not define \( X \). Thus, the answer to part c is: **C** \( \bar{X}= \) the average length, in minutes, of the songs from a sample of seven albums from the collection.