Chapter 7: Central Limit Theorem (Mean)

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9 ppm and standard deviation 1.4 ppm. 35 randomly selected large cities are studied. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. What is the probability that one randomly selected city's waterway will have less than 8.4 ppm pollutants? d. For the 35 cities, find the probability that the average amount of pollutants is less than 8.4 ppm. e. For part d), is the assumption that the distribution is normal necessary? f. Q1 = Q2 = IQR =

A. (9 , 1.4) B. (9, 0.2366) C. 0.3341 D. 0.0056 E. NO F. 8.8404 , 9.1596, .3192

Which of the following statements are true? I. The sampling distribution of ¯x¯ has standard deviation σ /√n even if the population is not normally distributed. II. The sampling distribution of ¯x¯ is normal if the population has a normal distribution. III. When n is large, the sampling distribution of ¯x¯ is approximately normal even if the the population is not normally distributed.

I, II, and III

When should you use t scores? I. The population is normally distributed and the population standard deviation, σ, is known, regardless of the sample size. II. The sample size is less than 30, and the population is not normally distributed. III. The sample size is less than 30, the population is normally distributed and the population standard deviation is NOT known.

III

Suppose the age that children learn to walk is normally distributed with mean 13 months and standard deviation 2.3 month. 35 randomly selected people were asked what age they learned to walk. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. What is the probability that one randomly selected person learned to walk when the person was between 12.7 and 13.5 months old? d. For the 35 people, find the probability that the average age that they learned to walk is between 12.7 and 13.5 months old. e. For part d), is the assumption that the distribution is normal necessary? f. Q1 = Q3 = IQR =

a. ( 13 , 2.3) b. (13 , 0.3888) c. 0.1379 d. 0.6806 e. No f. 12.7378 , 13.2622, 0.5245

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 147.6-cm and a standard deviation of 0.9-cm. For shipment, 15 steel rods are bundled together. Round all answers to four decimal places if necessary. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. For a single randomly selected steel rod, find the probability that the length is between 147.5-cm and 147.6-cm. d. For a bundled of 15 rods, find the probability that the average length is between 147.5-cm and 147.6-cm. e. For part d), is the assumption of normal necessary?

a. (147.6 , 0.9) b. (147.6 , 0.2324) c. 0.0442 d. 0.1665 e. yes

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 $159,000. Assume the standard deviation is $32,000. Suppose you take a simple random sample of 44 graduates. Round all answers to four decimal places if necessary. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. For a single randomly selected graduate, find the probability that her salary is between $158,164 and $163,776. d. For a simple random sample of 44 graduates, find the probability that the average salary is between $158,164 and $163,776. e. For part d), is the assumption of normal necessary?

a. (159000 , 32000) b. ( 159000 , 4824.1815) c. 0.0697 d. 0.4077 e. No

The average amount of money spent for lunch per person in the college cafeteria is $5.71 and the standard deviation is $2. Suppose that 18 randomly selected lunch patrons are observed. Assume the distribution of money spent is normal, and round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. For a single randomly selected lunch patron, find the probability that this patron's lunch cost is between $5.1429 and $5.4886 d. For the group of 18 patrons, find the probability that the average lunch cost is between $5.1429 and $5.4886. e. For part d), is the assumption that the distribution is normal necessary?

a. (5.71 , 2) b. (5.71, .4714) c. 0.0675 d. 0.2048 e. Yes

The amount of syrup that people put on their pancakes is normally distributed with mean 57 mL and standard deviation 7 mL. Suppose that 40 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. If a single randomly selected individual is observed, find the probability that this person consumes is between 58.1 mL and 59 mL. d. For the group of 40 pancake eaters, find the probability that the average amount of syrup is between 58.1 mL and 59 mL. e. For part d), is the assumption that the distribution is normal necessary?

a. (57, 7) b. (57, 1.1068) c. 0.05 d. 0.1248 e. No

Which of the following is true about sampling distributions? Sampling distribution of the mean is always right skewed since means cannot be smaller than 0. Sampling distributions are always nearly normal. Shape of the sampling distribution is always the same shape as the population distribution, no matter what the sample size is. Sampling distributions get closer to normality as the sample size increases.

Sampling distributions get closer to normality as the sample size increases.

Which of the following is true about the sampling distribution of means? Sampling distribution of the mean is always right skewed since means cannot be smaller than 0. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is. Sampling distributions of means get closer to normality as the sample size increases. Sampling distributions of means are always nearly normal.

Sampling distributions of means get closer to normality as the sample size increases.

The average number of miles (in thousands) that a car's tire will function before needing replacement is 73 and the standard deviation is 16. Suppose that 43 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X ~ N( , ) b. What is the distribution of ¯x¯? ¯x¯ ~ N( , ) c. 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 69.9 and 73.3. d. For the 43 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 69.9 and 73.3. e. For part d), is the assumption that the distribution is normal necessary?

a. (73 , 16) b. (73 , 2.4400) c. 0.0843 d. 0.447 e. No