ck songs, 40 jazz songs, 25 country songs, 30 hip - c songs. which statements are true? if crystal plays…

ck songs, 40 jazz songs, 25 country songs, 30 hip - c songs. which statements are true? if crystal plays songs at random, she should expect over half of them to be jazz or classical music songs. if crystal plays 100 songs at random, she should expect about 30 rock songs. if crystal plays songs at random, she should expect twice as many rock songs as hip hop songs. if crystal plays 50 songs at random, she should expect about 6 country songs.

ck songs, 40 jazz songs, 25 country songs, 30 hip - c songs. which statements are true? if crystal plays songs at random, she should expect over half of them to be jazz or classical music songs. if crystal plays 100 songs at random, she should expect about 30 rock songs. if crystal plays songs at random, she should expect twice as many rock songs as hip hop songs. if crystal plays 50 songs at random, she should expect about 6 country songs.

Answer

  1. First, assume the number of rock songs is (x), jazz songs is (40), country songs is (25), hip - hop songs is (30), and classical songs is (y). Let's assume we know the total number of songs (N=x + 40+25 + 30+y). For the sake of simplicity, if we just consider the proportion based on the given non - classical song numbers, the total number of non - classical songs (T=x + 40+25 + 30).
    • For the first statement:
      • We don't know the number of classical songs. If we only consider the non - classical songs, the number of jazz and classical (unknown) songs. Without knowing the number of classical songs, we can't say that over half of the songs will be jazz or classical. Let's assume there are no classical songs for now. The total number of non - classical songs is (x + 40+25 + 30). If (x) is large enough, this statement can be false.
    • For the second statement:
      • We need to know the proportion of rock songs. But we don't know the total number of songs. If we assume we are making an estimate based on a sample of 100 songs, we need to know the proportion of rock songs in the whole collection. Since we don't know the total number of songs and the number of rock songs (x), we can't say she should expect about 30 rock songs out of 100.
    • For the third statement:
      • We don't know the number of rock songs (x). If (x = 60) (twice of 30, the number of hip - hop songs), then if Crystal plays songs at random, she should expect twice as many rock songs as hip - hop songs. But if (x\neq60), this statement is false.
    • For the fourth statement:
      • The proportion of country songs in the collection is (p=\frac{25}{x + 40+25 + 30}). If she plays (n = 50) songs at random, the expected number of country songs (E=n\times p=50\times\frac{25}{x + 40+25 + 30}). If we assume the total number of non - classical songs (x+40 + 25+30=125) (for simplicity, assume (x = 30)), then (p=\frac{25}{125}=\frac{1}{5}). And (E = 50\times\frac{1}{5}=10\neq6).

Since the problem is incomplete (we don't know the number of rock and classical songs), we can't determine which statements are true. But if we assume some values for the sake of analysis:

Explanation:

Step1: Calculate proportion of each song type

Let's assume the total number of non - classical songs (N=30 + 40+25 + 30=125) (assuming 30 rock songs for illustration). The proportion of country songs (p_{country}=\frac{25}{125}=\frac{1}{5}), proportion of hip - hop songs (p_{hip - hop}=\frac{30}{125}=\frac{6}{25}), proportion of jazz songs (p_{jazz}=\frac{40}{125}=\frac{8}{25}).

Step2: Calculate expected number of songs for each case

For the fourth statement, if (n = 50) songs are played, the expected number of country songs (E=n\times p_{country}=50\times\frac{1}{5}=10\neq6). We still can't fully determine the other statements without more information. But if we assume the number of rock songs is 60: The proportion of rock songs (p_{rock}=\frac{60}{60 + 40+25 + 30}=\frac{60}{155}=\frac{12}{31}), proportion of hip - hop songs (p_{hip - hop}=\frac{30}{155}=\frac{6}{31}), and (p_{rock} = 2p_{hip - hop}).

Answer:

We need more information to accurately determine which statements are true. If we assume the number of rock songs is 60, the statement "If Crystal plays songs at random, she should expect twice as many rock songs as hip - hop songs" is true. But without knowing the full composition of the song collection (number of rock and classical songs), we can't be sure.