Practice Problems

1. The effective life of a component used in a jet-turbine aircraft engine is a random variable with mean 5000 hours and standard deviation 40 hours. The distribution of effective life is fairly close to a normal distribution. The engine manufacturer introduces an improvement into the manufacturing process for this component that increases the mean life to 5050 hours and decreases the standard deviation to 30 hours. Suppose that a random sample of n_{1} =16 components is selected from the “old” process and a random sample of n_{2} = 25 components is selected from the “improved” process. What is the probability that the difference in the two sample means is at least 25 hours? Assume that the old and improved processes can be regarded as independent populations.

2. The life in hours of a 75-watt light bulb is known to be approximately normally distributed, with standard deviation of 25 hours. A random sample of 20 bulbs as a mean life of 1014 hours.

- Construct a 95 percent two-sided confidence interval on the mean life.
- Construct a 95 percent lower-confidence interval on the mean life.

3. A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance of 1000 (psi)^{2}. A random sample of 12 specimens has a mean compressive strength of 3250 psi.

- Construct a 95 percent two-sided confidence interval on mean compressive strength.
- Construct a 99 percent two-sided confidence interval on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (a).

4. Suppose that in problem #2 we wanted to be 95 percent confident that the error in estimating the mean life is less than five hours. What sample size should be used?

5. Suppose that in problem #2 we wanted the total width of the confidence interval on mean life to be eight hours. What sample size should be used?

6. Suppose that in problem #3 it is desired to estimate the compressive strength with an error that is less than 15 psi. What sample size is required?

7. A machine produces metal rods in an automobile suspension system. A random sample below. Assume that rod diameter is normally distributed. Construct a 95 percent two-sided confidence interval on the mean rod diameter.

8.24 mm 8.21 8.23 8.25 8.26 | 8.23 mm 8.2 8.26 8.19 8.35 | 8.20 mm 8.28 8.24 8.25 8.24 |

8. Random samples of size 20 were drawn from two independent normal populations. The sample means were 22.0 and 21.5, while the standard deviations were 1.5 and 1.8. Construct a 95 percent two-sided confidence interval on μ _{1} – μ _{2}.

9. A computer scientist is investigating the usefulness of two different design languages in improving programming tasks. Twelve expert programmers, familiar with both languages, are asked to code a standard function in both languages, and the time in minutes is recorded. The data are shown below:

Time | ||

Programmer | Design Language 1 | Design Language 2 |

1 | 17 | 18 |

2 | 16 | 14 |

3 | 21 | 19 |

4 | 14 | 11 |

5 | 18 | 23 |

6 | 24 | 21 |

7 | 16 | 10 |

8 | 14 | 13 |

9 | 21 | 19 |

10 | 23 | 24 |

11 | 13 | 15 |

12 | 18 | 20 |

Find a 95 percent confidence interval on the difference in mean coding times. Is there any indication that one design language is preferable?

10. An automatic filling machine is used to fill bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of fill volume of s^{2} = 0.0153 (fluid ounces)^{2}. If the variance of fill volume is too large, an unacceptable proportion of bottles will be under- or overfilled. We will assume that the fill volume is approximately normally distributed. Find a 95% upper-confidence interval on the population variance:

11. A random sample of 50 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 18 of these helmets some damage was observed.

(a) Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test.

(b) Using the point estimate of p obtained from the preliminary sample of 50 helmets, how many helmets must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02?