The average playing time of compact discs in a large collection is 35 minutes, a

Question

The average playing time of compact discs in a large collection is 35 minutes, and the standard deviation is 7 minuzes s 1 standard deviation above the mean? 1 standard deviation below the mean? what values are 2 standard deviations sway from the mean? 1 standard deviation above the mearn 1 standard deviation below the mean 2 standard deviations above the mean 2 standard deviations below the mean (b) Without assuming anything about the distribution of times, at least what percentage of the times are between 2t and 49 minutes? (Round the answer to the nearest whole number.) At least (c) without assuming No more than anything about the distrihution of times, what can be said about the percentage of timas that are either less than 14 minutes or greater than 56 mirutes? (Round the answer to the nearest whole nember.] that the distribution of times is normal, about what percentage of times are between 21 and 49 minutes? (Round the answers to tno decimal places, needed.) OAu Less than 14 min or greater than 56 min? Less than 14 min?

Explanation / Answer

a)1) 1 standard deviation above the mean = 35 + 7 = 42

    2) 1 standard deviation below the mean = 35 - 7 = 28

    3) 2 standard deviation above the mean = 35 + 2 * 7 = 49

    4) 2 standard deviation below the mean = 35 - 2 * 7 = 21

b) 21 is two standard deviation below the mean and 49 is two standard deviation above the mean.

According to emperical rule about 95% of the data lies between two standard from the mean

P(21 < X < 49) = 95%

c) 14 is three standard deviation below the mean and 56 is two standard deviation above the mean

According to emperical rule about 99.7% of the data lies between three standard from the mean.

So the percentage of times less than 14 or greater than 56 = 1 - 0.997 = 0.003 = 0.3%

d) P(21 < X < 49)

= P((21 - mean)/sd < (X - mean)/sd < (49 - mean)/sd)

= P((21 - 35)/7 < Z < (49 - 35)/7)

= P(-2 < Z < 2)

= P(Z < 2) - P(Z < -2)

= 0.9772 - 0.0228

= 0.9544 = 95.44%

1 - ( P(14 < X < 56))

= 1 - P((14 - mean)/sd < (X - mean)/sd > (56 - mean)/sd))

= 1 - (P(14 - 35)/7) < Z < (56 - 35)/7))

= 1 - P(-3 < Z < 3)

= 1 - (P(Z < 3) - P(Z < -3))

= 1 - (0.9987 - 0.0013)

= 0.0026 = 0.26%

The percentage of time lies beteen less than 14 and above 56 is 0.26%

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