Nine Practice Problems -- Normal Distribution
1. Replacement times for CD players are normally distributed with a mean of 7.1 years and a standard deviation of 1.4 years (data from Consumer Reports). What is the probability that a randomly-selected CD player will have to be replaced in 8 years or less?
2.Replacement times for CD players are normally distributed with a mean of 7.1 years and a standard deviation of 1.4 years (data from Consumer Reports). If you are the manufacturer and want to provide a warranty such that 98% of the players need replacement after the warranty expires, how long should the warranty period be?
3.A mile-runner’s times for the mile are normally distributed with a mean of 4 min. 3 sec. (This would have to be expressed in decimal minutes -- 4.05 minutes), and a standard deviation of 2 seconds (0.0333333··· minutes (the three dots indicate a repeating decimal)). What is the probability that on a given run, the time will be 4 minutes or less?
4.A mile-runner’s times for the mile are normally distributed with a mean of 4 min. 3 sec. (This would have to be expressed in decimal minutes -- 4.05 minutes), and a standard deviation of 2 seconds (0.0333333··· minutes (the underline indicates a repeating decimal)). What does the mean have to be for a 0.20 probability of the time being 4 minutes or less?
5.A machine fills 24-ounce (according to the label) boxes with cereal. The amount deposited into the box is normally distributed with a standard deviation of 0.25 ounce. What does the mean have to be in order for 99.5% of the boxes to contain 24 ounces or more of cereal?
6.A machine fills 24-ounce (according to the label) boxes with cereal. The amount deposited into the box is normally distributed with a standard deviation of 0.20 ounce. What does the mean have to be in order for only 0.2% of the boxes to contain more than 24.5 ounces or more of cereal?
7.A student gets a 70 on a test where the mean score was 64. What does the standard deviation have to be in order for the student to be in the 95th percentile?
8.A machine fills 24-ounce (according to the label) boxes with cereal. The amount deposited into the box is normally distributed with a mean of 24.8 ounces. What does the standard deviation have to be in order for 96% of the boxes to contain 24 ounces or more?
9. On a standardized test, scores are normally distributed with a mean of 400 and a standard deviation of 80. What score must one have to be in the 80th percentile?
The blue italic number indicates the one computed by one of the four formulas.
zxμσAnswer
1.0.64285787.11.40.739842
2.-2.053754.224757.11.44.22475
3.-1.5000044.050.03333330.0668072
4.-0.84162144.028050.03333334.02805*
* 4.02805 minutes = 4 minutes 1.683 seconds
5.-2.575832424.64400.2524.6440
6.2.8781624.523.92440.2023.9244
7.1.6448570643.647743.64774
8.-1.750692424.80.4569640.456964
9.0.841621467.33040080467.330