Exam 8: Sampling Variability Sampling

In a balanced ecosystem, all animals are subject to predators. Bees have been observed avoiding flowers where spiders have been present, but it is not known how they detect the spider’s presence. A series of studies involving Italian honeybees and crab spiders was conducted to address this question. One study looked at flowers that had a dead bee on its petals. In a sample of 128 bee flights to flowers with dead bees, 83 resulted in the bee inspecting the flower, and 38 of these inspections resulted in the bee “accepting” the flower. -Suppose you are interested in learning about the value of p, the proportion of allflowers with bee remains that would be inspected. This proportion can be estimatedusing the sample proportion, $\hat{ p}$ . What is the sample proportion for this sample?

Consider sampling from a population whose proportion of successes is p = 0.5. Asthe sample size, n, increases, some characteristics of the sampling distribution ofchange. Which of the following characteristics will change as n increases, and whatis the nature of the change? a) The mean of the sampling distribution of $\hat{p}$ b) The standard deviation of the sampling distribution of $\hat{p}$ c) The shape of the sampling distribution of $\hat{p}$

The sampling variability of $\hat{ p}$ is caused by the population proportionvarying from sample to sample.

The characteristics of sampling distributions that we are interested in are: center,shape, and spread. In a few sentences, describe these characteristics for the samplingdistribution of $\hat{ p}$ .

In a balanced ecosystem, all animals are subject to predators. Bees have been observed avoiding flowers where spiders have been present, but it is not known how they detect the spider’s presence. A series of studies involving Italian honeybees and crab spiders was conducted to address this question. One study looked at flowers that had a dead bee on its petals. In a sample of 128 bee flights to flowers with dead bees, 83 resulted in the bee inspecting the flower, and 38 of these inspections resulted in the bee “accepting” the flower. -In prior studies with different species of bees, about 70% of flowers containing deadbees had resulted in acceptance of the flower. Based on the given data, is itreasonable to conclude that the proportion of landings of Apis mellifera is differentfrom 0.70?

The medical field of pediatrics has consistently attracted a large number of women. The challenge of balancing a personal and professional life is of particular concern to women, and one possible solution is to take a part-time position. A survey of female pediatric residents inquired about the barriers to seeking part-time or reduced-hours employment in the field of pediatrics. Out of 235 women who responded, 153 cited loss of income and 101 cited reduction in benefits as potential barriers. -Suppose you are interested in the value of p, the proportion of all women pediatricsresidents for whom loss of income would be a barrier to part-time work. Thisproportion can be estimated using the sample proportion, $\hat{p}$ . What is the sampleproportion for this sample?

The characteristics of sampling distributions that we are interested in are: center,shape, and spread. In a few sentences, describe these characteristics for the samplingdistribution of $\hat{ p}$ .

In a balanced ecosystem, all animals are subject to predators. Bees have been observed avoiding flowers where spiders have been present, but it is not known how they detect the spider’s presence. A series of studies involving Italian honeybees and crab spiders was conducted to address this question. One study looked at flowers that had a dead bee on its petals. In a sample of 128 bee flights to flowers with dead bees, 83 resulted in the bee inspecting the flower, and 38 of these inspections resulted in the bee “accepting” the flower. -Based on what you know about the sampling distribution of $\hat{ p}$ , is it reasonable tothink that this estimate is within 0.05 of the actual value of the population proportion? Explain why or why not.

The distribution of $\hat{ p}$ always has the same shape, approximately normal,irrespective of the value of p and the sample size.

A landlord is considering raising the monthly rent because of increased costs. He isconcerned about competition, and will only increase the rent if the current local cityvacancy rate is below 20%. Suppose the owner has decided to take a simple randomsample of apartments in the city and estimate the vacancy rate. He is considering asample size of n = 40. Suppose further that the actual vacancy rate is 0.30. a) Show that this sample size is large enough for the sampling distribution of $\hat{p}$ to bapproximately normal. b) What is the mean of the sampling distribution of $\hat{p}$ ? c) What is the standard deviation of the sampling distribution of $\hat{p}$ ? d) What is the approximate probability that a sample proportion, $\hat{p}$ , would be lessthan 0.20 even though the population proportion is 0.30?

The medical field of pediatrics has consistently attracted a large number of women. The challenge of balancing a personal and professional life is of particular concern to women, and one possible solution is to take a part-time position. A survey of female pediatric residents inquired about the barriers to seeking part-time or reduced-hours employment in the field of pediatrics. Out of 235 women who responded, 153 cited loss of income and 101 cited reduction in benefits as potential barriers. -Based on what you know about the sampling distribution of $\hat{ p}$ , is it reasonable tothink that this estimate is within 0.05 of the actual value of the population proportion? Explain why or why not.

The principal at Thomas Jefferson High School has been asked to estimate theproportion of students at the school who drive to school and use the school parkinglot. He takes a random sample of size n = 32 students and calculates a sampleproportion, $\hat{ p}$ = 0.8. "Now," he exclaims, "since my sample size is greater than 30,the sampling distribution of the sample proportion is approximately normal." Writea short paragraph that explains why his statement is not correct.

A statistic is a characteristic of a population.

For a given population proportion, the sampling distribution of $\hat{ p}$ tends tobe more spread out for larger sample sizes than for smaller sample sizes.

For n sufficiently large, and p sufficiently close to 0.5, the distribution of is approximately a standard normal distribution.

The medical field of pediatrics has consistently attracted a large number of women. The challenge of balancing a personal and professional life is of particular concern to women, and one possible solution is to take a part-time position. A survey of female pediatric residents inquired about the barriers to seeking part-time or reduced-hours employment in the field of pediatrics. Out of 235 women who responded, 153 cited loss of income and 101 cited reduction in benefits as potential barriers. -Based on past experience researchers had theorized that about 0.56 of respondentswould cite reduction of benefits as a potential barrier. Based on the given data, is itreasonable to conclude that this proportion for all women pediatrics residents differsfrom 0.56?

For a given sample size, the sampling distribution of $\hat{ p}$ tends to be morespread out as p gets closer to 1.0.

If p = 0.90, a sample size of n =10 is large enough for the samplingdistribution of $\hat{ p}$ to be well approximated by a normal distribution.

As n increases, the mean of the sampling distribution of $\hat{ p}$ gets closerto p .

The principal at George Washington High School has been asked to estimate theproportion of students at the school who drive to school and use the school parkinglot. He takes a random sample of size n = 40 students and calculates a sampleproportion, $\hat{ p}$ = 0.8. "Now," he exclaims, "since my sample size is pretty large, thesampling distribution of the sample proportion is approximately normal." W rite ashort paragraph that explains why his statement is not correct.