Question

the following table shows a random sample of the lengths of songs in a playlist.
lengths of songs (minutes:seconds)
3:44 3:39 3:48 3:19 3:16 3:35
2:29 3:00 3:45 3:48 4:03 3:29
(a) find the mean and sample standard deviation of the song lengths, in seconds. round each result to the nearest second.
mean sec
sample standard deviation sec
(b) are there any song lengths in the data set that do not lie within 1 standard deviation of the mean? if so, list the times in seconds. (if an answer does not exist, enter dne. enter your answers as a comma - separated list.)

Explanation:

Step1: Convert times to seconds

• \(3:44 = 3\times60 + 44=224\)
• \(3:39 = 3\times60+39 = 219\)
• \(3:48=3\times60 + 48=228\)
• \(3:19=3\times60+19 = 199\)
• \(3:16=3\times60+16 = 196\)
• \(3:35=3\times60+35 = 215\)
• \(2:29=2\times60+29 = 149\)
• \(3:00=3\times60+0 = 180\)
• \(3:45=3\times60+45 = 225\)
• \(3:48=228\)
• \(4:03=4\times60+3 = 243\)
• \(3:29=3\times60+29 = 209\)

Step2: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 12\) and \(\sum_{i=1}^{12}x_{i}=224 + 219+228+199+196+215+149+180+225+228+243+209\)
\(\sum_{i=1}^{12}x_{i}=2664\)
\(\bar{x}=\frac{2664}{12}=222\)

Step3: Calculate the sample variance \(s^{2}\)

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\((224 - 222)^{2}=4\), \((219-222)^{2}=9\), \((228 - 222)^{2}=36\), \((199-222)^{2}=529\), \((196-222)^{2}=676\), \((215-222)^{2}=49\), \((149-222)^{2}=5329\), \((180-222)^{2}=1764\), \((225-222)^{2}=9\), \((228-222)^{2}=36\), \((243-222)^{2}=441\), \((209-222)^{2}=169\)
\(\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}=4+9+36+529+676+49+5329+1764+9+36+441+169=9041\)
\(s^{2}=\frac{9041}{11}\approx821.91\)

Step4: Calculate the sample standard deviation \(s\)

\(s=\sqrt{s^{2}}=\sqrt{821.91}\approx29\)

Step5: Find the range for 1 - standard deviation

The range is \(\bar{x}-s\) to \(\bar{x}+s\), i.e., \(222-29 = 193\) to \(222 + 29=251\)
Check each value:
\(149<193\)

Answer:

a. mean: \(222\) sec, sample standard deviation: \(29\) sec
b. \(149\)