Question

The life of a coffee pot is approximately normally distributed with a mean of 5 years and standard deviation of 1.1 years. The manufacturer will repair or replace any defective coffee pot free of charge (including labor) while the pot is under warranty. For how many years should the company guarantee its coffee pots, if the manufacturer does not wish to replace more than 5% of them? The playing times for cassette tapes manufactured by the Econo corporation are normally distributed with a mean of 46.4 minutes and a standard deviation of 2.5 minutes. a) What is the probability that a randomly selected Econo cassette tape has a playing time of more than 48 minutes? b) What is the probability that a randomly selected Econo cassette tape has a playing time that is between 42 and 45 minutes?

Answer 1

The warranty period for a coffee pot should be determined by calculating the Z-score corresponding to the 95th percentile and solving using the mean and standard deviation. The probabilities for the cassette tapes' playing time are found by converting to Z-scores and calculating the area under the normal curve using a normal table or calculator.

Step-by-step explanation:

The student's query pertains to probability calculations involving normally distributed data. In the first scenario, we are told that the life of a coffee pot is normally distributed with a mean of 5 years and a standard deviation of 1.1 years. To determine the warranty period such that no more than 5% of the coffee pots need replacement, we must find the value on the lower tail of the distribution that leaves 5% in the tail. We use the Z-score formula to find this value, which involves finding the Z that corresponds to the 95th percentile and then solving for the warranty period using the mean and standard deviation.

For the Econo corporation's cassette tapes, we perform two separate probability calculations:

1. The probability of a tape having a playtime of more than 48 minutes is computed by finding the area under the normal curve to the right of 48 minutes.
2. The probability of a tape having a playtime between 42 and 45 minutes is the area under the curve between these two values.

Both calculations involve converting the minutes to Z-scores and then using a standard normal table or a calculator function like normalcdf to find the probabilities.