Unit 5: Sampling Distributions Review

The central limit theorem is important in statistics because it allows us to use the Normal distribution to find probabilities involving the sample mean

IF the sample size is reasonably large (equal to or greater than a sample size of 30)

Answer the question in the image

Mean = 2, SD = 0.85

Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with a mean 515 and standard deviation 114. You choose an SRS of 100 students and average their SAT math scores. Suppose that you do this many, many times. Which of the following are the mean and standard deviation of the sampling distribution of x-bar?

Mean = 515, Standard Deviation 114/110

Yuki is a candidate is running for office, and she wants to know how much support she has in two different districts. Yuki doesn't know it, but 45% of the 8,000 voters in District A support her, while 40 percent of the 6,500 voters in District B support her. Yuki hires a polling firm to take separate random samples of 100 voters from each district. The firm will then look at the difference between the proportions of voters who support her in each sample. What are the mean and standard deviation of the sampling distribution of p-hat A - p-hat B?

Mean: 0.05 Standard Deviation: 0.070

Answer the question in the image.

0.05

The life span of a battery is the amount of time the battery will last. The distribution of life span for a certain type of battery is approximately normal with mean 2.5 hours and standard deviation 0.25 hour. Suppose one battery will be selected at random. Which of the following is closest to the probability that the selected battery will have a life span of at most 2.1 hours?

0.055

A river runs through a certain city and divides the city into two parts, west and east. The population proportion of residents in the west who are in favor of building a new bridge across the river is known to be pW=0.30. The population proportion of residents in the east who are in favor of building a new bridge across the river is known to be pE=0.20. For sample sizes of size 50 what is the mean of the sampling distribution of the differences in the sample proportions?

0.10

In a certain region of the country, the proportion of the population with blue eyes is currently 17 percent. A random sample of 100 people will be selected from the population. What is the mean of the sampling distribution of the sample proportion of people with blue eyes for samples of size 100 ?

0.17

Suppose we select an SRS of size n=100 from a large population having proportion p of successes. Let p-hat be the proportion of success in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of p-hat?

0.85

The distribution of the amount of water used per wash cycle, in gallons, by a particular brand of dishwasher during a wash cycle is approximately normal with mean 4 gallons and standard deviation 0.2 gallon. Which of the following is closest to the probability that more than 3.75 gallons are used for any wash cycle?

0.89

A study of voting chose 663 registered voters at random shortly after an election. Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election. Which of the following statements is true about the boldface numbers?

72% is a statistic and 56% is a parameter

The distribution of the number of siblings for students at a large high school is skewed to the right with mean 1.8 siblings and standard deviation 0.7 sibling. A random sample of 100 students from the high school will be selected, and the mean number of siblings in the sample will be calculated. Which of the following describes the sampling distribution of the sample mean for samples of size 100 ?

Approximately normal with a standard deviation LESS than 0.7 sibling.

The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85. Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?

Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.

The mean age of the employees at a large corporation is 35.2 years, and the standard deviation is 9.5 years. A random sample of 4 employees will be selected. What are the mean and standard deviation of the sampling distribution of the sample mean for samples of size 4

The mean is 35.2 and the standard deviation is 9.5/2

A survey of 100 randomly selected dentists in the state of Ohio results in 78% who would recommend the use of a certain toothpaste. The population proportion is known to be p=0.72. For samples of size 100, which of the following best interprets the mean of the sampling distribution of sample proportion of dentists in the state of Ohio who would recommend the use of a certain toothpaste?

The mean of all sample proportions from all random samples of 100 dentists in the state of Ohio is equal to 0.72.

A sports magazine reports that the mean number of hot dogs sold by hot dog vendors at a certain sporting event is equal to 150. A random sample of 50 hot dog vendors was selected, and the mean number of hot dogs sold by the vendors at the sporting event was 140. For samples of size 50, which of the following is true about the sampling distribution of the sample mean number of hot dogs sold by hot dog vendors at the sporting event?

The mean of the sampling distribution of the sample mean is 150 hot dogs.

According to a recent survey, 47.9 percent of housing units in a large city are rentals. A sample of 210 housing units will be randomly selected. Which of the following must be true for the sampling distribution of the sample proportion of housing units in the large city that are rentals to be approximately normal?

The values of 210(0.479) and 210(0.521)

The student newspaper at a large university asks an SRS of 250 undergraduates, "Do you favor eliminating the carnival from the term-end celebration?" All in all, 150 of the 250 are in favor. Suppose that (unknown to you) 55% of all undergraduates favor eliminating the carnival. If you took a very large number of SRSs of size n=250 from this population, the sampling distribution of the sample proportion p-hat would be

approximately Normal with a mean of 0.55 and standard deviation of 0.03