Exam 8: Sampling Variability and Sampling Distributions

The principal at John F. Kennedy High School has been asked to provide the average number of classes taken by the students at KHS. Since the computer system is down, she takes her alphabetized list of students, randomly selects 50 students, determines the number of classes each of the 50 selected students is taking, and calculates = 5.4. She then reports to the PTA "Since I took a large random sample, the population mean number of classes taken by the students at KHS is 5.4." Write a short paragraph to send to her that explains why her statement is not correct.

Which of the following symbols is used to indicate a population proportion? ​

The mean of the sampling distribution of is p no matter how large n is.

Some biologists believe the evolution of handedness is linked to complex behaviors such as tool-use. Under this theory, handedness would be genetically passed on from parents to children. That is, left-handed parents would be more likely to have left-handed children than right-handed parents. An alternate theory asserts that handedness should be random, with left- and right-handedness equally likely. In a recent study using a simple random sample of n = 76 right-handed parents, 50 of the children born were right-handed. ( = 0.658.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p = 0.50. a)Show that it is reasonable to approximate the sampling distribution of p using a normal distribution. b)Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least = 0.658?

Consider the following "population": {1, 1, 3, 4}. Suppose that a random sample of size n = 2 is to be selected without replacement from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of . (Display it in table form.)

One method for estimating the availability of office space in large cities is to conduct a random sample of offices, and calculate the proportion of offices currently being used. Suppose that real estate agents believe that p = 0.70 of all offices are currently occupied, and decide to take a sample to assess their belief. They are considering a sample size of n = 40. a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of . b)What is the mean of the sampling distribution of if the real estate agents are correct? c)What is the standard deviation of the sampling distribution of if the real estate agents are correct? d)If the real estate agents are correct, what is the probability that a sample proportion, , would differ from p = 0.70 by as much as 0.05?

A statistic is a characteristic of the population.

Consider sampling from a population whose proportion of successes is p = 0.1. As the sample size, n, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as n increases, and what is the nature of the change? a)The mean of the sampling distribution of b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of

The standard deviation of the distribution of decreases as n increases.

How are the quantities, p and , related?

If p = 0.90, a sample size of n = 10 is large enough for the sampling distribution to be well approximated by a normal distribution.

A sampling distribution of can be regarded as approximately normal when what conditions are met? ​

The first large-scale study of the human sex ratio involved over 6,000 families each having 12 children. (This was done in 19th Century Germany--large families were more common then and there.) 52% of the children they observed were boys. Suppose that 21st Century researchers wish to replicate this observational study. In order to see if this proportion of boys might have changed in the intervening years, suppose the researchers track down 50 families with 12 children. From these 600 children, a random sample of 50 children is taken. 30 of the 50 children were boys ( = 0.6.)a)Show that it is reasonable to approximate the sampling distribution of using a normal distribution. b)If the modern true population proportion of newborn boys is p = 0.52, what is the probability of observing a sample proportion of at least = 0.6?

Consider sampling from a population whose proportion of successes is p = 0.5. As the sample size, n, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as n increases, and what is the nature of the change? a)The mean of the sampling distribution of b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of

Suppose that the actual proportion of people at a particular company who use their bicycles to travel to job is 0.35. In a study of bicycle parking places needs, administration would like to estimate this proportion. They plan to take a random sample of 50 employees and use the sample proportion who use bicycles, , as an estimate of the population proportion. Estimate the standard deviation of the sample distribution. Suppose that another sample of different size has been selected, and the standard deviation of this new sample is . Would you expect more or less sample-to-sample variability in the sample proportions than for when ? Is the sample size that resulted in larger or smaller than 50?

Suppose that 8% of the customers of a cable television company watch the Jewelry Network channel at least once a week. The cable company does not know the actual proportion of all customers who watch the channel at least once a week and is trying to decide whether to replace this channel with a new local station. The company plans to take a random sample of 150 customers and to use as an estimate of the population proportion. ​ Determine the standard deviation of . ​

When the sample size is "large enough," the statistic p has an approximately normal sampling distribution. How does one determine if a sample size is large enough?

Suppose that a particular candidate for public office is favored by 43% of all registered voters in the district A polling organization will take a random sample of 50 of these voters and will use , the sample proportion, to estimate p. ​ Determine the standard deviation of . ​

Determine the mean and standard deviation of the sampling distribution of when n = 100 and p = 0.65.

One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that π = 0.453 of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska. a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of . b)What is the mean of the sampling distribution of ? c)What is the standard deviation of the sampling distribution of ?