8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
2:16 minutesProblem 6.3.1a
Textbook Question
Fatal Car Crashes There are about 15,000 car crashes each day in the United States, and the proportion of car crashes that are fatal is 0.00559 (based on data from the National Highway Traffic Safety Administration). Assume that each day, 1000 car crashes are randomly selected and the proportion of fatal car crashes is recorded.
a. What do you know about the mean of the sample proportions?
Verified step by step guidance1
The mean of the sample proportions is a measure of the central tendency of the sample proportions. According to the Central Limit Theorem, the mean of the sample proportions is equal to the population proportion (p).
In this case, the population proportion (p) is given as 0.00559, which represents the proportion of car crashes that are fatal.
Thus, the mean of the sample proportions (denoted as μ_p̂) is equal to the population proportion: μ_p̂ = p.
This result holds because the sampling distribution of the sample proportion is centered around the true population proportion when the samples are randomly selected.
To summarize, the mean of the sample proportions is μ_p̂ = 0.00559, which is the same as the population proportion.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Proportion
The sample proportion is the ratio of the number of successes (in this case, fatal car crashes) to the total number of observations in a sample. It is denoted as p̂ and provides an estimate of the population proportion. Understanding sample proportions is crucial for making inferences about the population based on sample data.
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Sampling Distribution of Sample Proportion
Mean of Sample Proportions
The mean of sample proportions refers to the expected value of the sample proportion across many samples. According to the Central Limit Theorem, this mean will be equal to the true population proportion (0.00559 in this case) when the sample size is sufficiently large, allowing for reliable estimates of the population parameter.
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Central Limit Theorem
The Central Limit Theorem states that the distribution of the sample means (or sample proportions) will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is fundamental in statistics as it justifies the use of normal probability models for inference when dealing with large samples.
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