Central Limit Theorem (Mean)

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 185.4-cm and a standard deviation of 1-cm. For shipment, 49 steel rods are bundled together. Round all answers to four decimal places if necessary. 

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

· ~ N(,) 

· For a single randomly selected steel rod, find the probability that the length is between 185.3-cm and 185.4-cm. 

· For a bundled of 49 rods, find the probability that the average length is between 185.3-cm and 185.4-cm. 

· For part d), is the assumption of normal necessary? NoYes

The average number of miles (in thousands) that a car's tire will function before needing replacement is 68 and the standard deviation is 15. Suppose that 50 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

1. ~ N(,)      
2. If a randomly selected individual tire is tested, find      the probability that the number of miles (in thousands) before it will      need replacement is between 68.7 and 70.8. 
3. For the 50 tires tested, find the probability that the      average miles (in thousands) before need of replacement is between 68.7      and 70.8. 
4. For part d), is the assumption that the distribution is      normal necessary? NoYes      

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.1 years and a standard deviation of 0.4 years. He then randomly selects records on 37 laptops sold in the past. Round the answers of following questions to 4 decimal places. 

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

· ~ N(,) 

· What is the probability that one randomly selected laptop is replaced more than 3.2 years? 

· For 37 laptops, find the probability that the average replacement time is more than 3.2 year. 

· For part d), is the assumption of normal necessary? YesNo

Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $170,000. Assume the standard deviation is $32,000. Suppose you take a simple random sample of 36 graduates. Round all answers to four decimal places if necessary. 

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

· ~ N(,) 

· For a single randomly selected graduate, find the probability that her salary is between $169,700 and $174,267. 

· For a simple random sample of 36 graduates, find the probability that the average salary is between $169,700 and $174,267. 

· For part d), is the assumption of normal necessary? YesNo

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 25 weeks. Assume that the length of unemployment is normally distributed with population mean of 25 weeks and the population standard deviation of 7 weeks. Suppose you would like to select a random sample of 17 unemployed individuals for a follow-up study. Round the answers of following questions to 4 decimal places. 

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

· ~ N(,) 

· What is the probability that one randomly selected individual found the next job is less than 24 weeks? 

· For 17 unemployed individuals, find the probability that the average time that they found the next job is less than 24 weeks. 

· For part d), is the assumption of normal necessary? NoYes

Suppose that the average number of Facebook friends users have is normally distributed with a mean of 126 and a standard deviation of about 55. Assume forty-nine individuals are randomly chosen. Answer the following questions. Round all answers to 4 decimal places where possible.

1. What is the distribution of 

? 

· ~ N(,) 

· For the group of 49, find the probability that the average number of friends is less than 136. 

· Find the first quartile for the average number of Facebook friends. 

· For part b), is the assumption that the distribution is normal necessary? YesNo

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9 ppm and standard deviation 1.7 ppm. 39 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.

1. What is the distribution of 

? 

· ~ N(,) 

· What is the distribution of ? 

1. ~ N(,)      
2. What is the probability that one randomly selected      city's waterway will have less than 9.2 ppm pollutants? 
3. For the 39 cities, find the probability that the      average amount of pollutants is less than 9.2 ppm. 
4. For part d), is the assumption that the distribution is      normal necessary? YesNo      
5. Find the IQR for the average of 39 cities.
        Q1 = ppm
        Q3 = ppm
        IQR: ppm