More Sampling Distributions

Recall: A statistic is a numerical value computed from a sample of observations. A statistic is a random variable. The distribution of a statistic is called its sampling distribution.

Sampling Distribution for a Sample Proportion Let p = population proportion of interest. Let then the sampling distribution of If the population is at least 10 times the sample size and is a normal distribution withand

Question: Suppose 20% of all New Hampshire voters favor John Edwards. Pollsters take a sample of 600 voters. Describe the mean and standard deviation of the sampling distribution of. Answer:

Question: Suppose 20% of all New Hampshire voters favor John Edwards. Determine the probability that the sample proportion will be between.18 and.22. Answer: Find Now Therefore: In conclusion, there is a 77.9% probability that the sample proportion for a sample of 600 voters will fall between.18 and.22, given p=.2. Recall that and so

Another Question: Suppose 20% of all New Hampshire voters favor John Edwards. Now determine the probability that the sample proportion will be between 5 percentage points of the mean. Answer: Find Now Therefore: In conclusion, there is a 99.8% probability that the sample proportion for a sample of 600 voters will fall between.15 and.25, given p=.2. Recall thatand so

Facts: The US Fish and Wildlife Service estimates that in the US between 4 and 5 million birds die annually when they strike communication towers. The American Bird Conservancy has investigated this assertion and reports that 53% of such birds killed are nocturnal migrants. Birds may be attracted to towers by their electromagnetic fields, or may just not see them in the dark.

The Rose-breasted Grosbeak migrates from the mountains of North Carolina and northern North America to Central and South America, where it winters. Like many songbirds much of its migration is at night. © Sven Halling

The Yellow-rumped Warbler breeds in northern North America and winters in the southern US and south to South America. They, also, are nocturnal migrants.

Question: Birds killed at certain towers are collected and sent for identification to the USFWS. If we take random samples of 100 such birds, describe the sampling distribution of the proportion that are nocturnal migrants.

Answer: First we check the assumptions and can assume that the population of all birds is far greater than ten times our sample size. The sampling distribution is N (.53,.04991). Next we check np and n(1-p). Both are greater than 10, so we meet the second assumption.

Question: What is the probability that the proportion of nocturnal migrants is between 45 and 61% for a random sample of 100 birds killed by communication towers? Answer: Find Now Therefore: In conclusion, there is a 89.1% probability that the sample proportion for a sample of 100 birds killed at communication towers will be made of 45 to 61% nocturnal migrants. Recall thatand so

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