Population Mean, Probability

Question

The population mean length of the single song records released by the Beatles during their highly distinguished careers is 205 seconds. The population standard deviation is 3 seconds, and the population is normally distributed. When a single song record is chosen at random, what is the probability that it will be:

A). P(201 secs < X < 202 secs)

B). P(X < 201 secs)

C). P(X > 206 secs)

D). P(X < 210 secs)

Answer 1

We will be computing the z-statistic and using the standard normal probability tables to find the probabilities.

For X = a single record, we standardize the value of X to a z statistic as follows:

z = X - population mean = X - 205

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population std dev 3

A). P(201 secs < X < 202 secs)

= P((201-205)/3 < Z < (202-205)/3)

= P(-1.33 < Z < -1)

because probabilities in the standard normal probability table are cumulative to the left

= P(Z < -1) - P(Z < -1.33)

= 0.1587 - 0.0918 = 0.0669

B). P(X < 201 secs)

= P(Z < (201-205)/3) = P(Z < -1.33) = 0.0918

C). P(X > 206 secs)

= P( Z > (206-205)/3) = P(Z > 0.333)

because the normal distribution is symmetric:

= P(Z < -0.333) = approximately 0.37

D). P(X < 210 secs)

= P (Z < (210-205)/3)

= P(Z < 1.67) = 0.9525