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Magazine Chapter 6: Problem 20
According to the U.S. Census, \(67.5 \%\) of the U.S. population were born in their state of residence. In a random sample of 200 Americans, what is the probability that fewer than 125 were born in their state of residence?
Short Answer
Expert verified
The probability is approximately 0.0618.
Step by step solution
01
Identify the Problem Type
This is a probability problem involving a sample proportion. We need to calculate the probability that a sample proportion is less than a certain value.
02
Define the Variables
Let us denote the probability that a person was born in their state of residence as \( p = 0.675 \). The sample size is \( n = 200 \). We aim to find the probability that fewer than 125 people in the sample were born in their state of residence.
03
Calculate the Sample Proportion
The sample proportion can be expressed as \( \hat{p} = \frac{X}{n} \), where \( X \) is the number of successes. We need to find \( P(X < 125) \).
04
Apply Normal Approximation
Since \( n \times p \) and \( n \times (1-p) \) are greater than 5, we can use the normal approximation of the binomial distribution. Calculate the mean \( \mu = n \times p = 200 \times 0.675 = 135 \) and standard deviation \( \sigma = \sqrt{n \times p \times (1-p)} = \sqrt{200 \times 0.675 \times 0.325} \approx 6.478 \).
05
Convert to Standard Normal Distribution
We convert \( X < 125 \) to a standard normal distribution using \( Z = \frac{X - \mu}{\sigma} \). Thus, \( Z = \frac{125 - 135}{6.478} \approx -1.542 \).
06
Find the Probability Using the Z-Table
Using the standard normal distribution table (Z-table), we find \( P(Z < -1.542) \). This probability is approximately 0.0618.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Sample Proportion
A sample proportion is an important concept in statistics, particularly in probability. It represents the fraction of a sample that possesses a certain characteristic. Calculating a sample proportion is crucial when determining probabilities involving samples.
- In the given scenario, the characteristic we're interested in is whether an individual was born in their state of residence.
- The sample proportion, denoted as \( \hat{p} \), is calculated using the formula \( \hat{p} = \frac{X}{n} \), where \( X \) is the number of successes (people born in their state of residence), and \( n \) is the sample size.
- This sample proportion helps us estimate the probability of specific outcomes within the large population based on our sample examination.
Exploring Normal Approximation
Normal approximation is a statistical technique used when working with binomial distributions that involve a large number of trials. This technique simplifies calculations by using the normal distribution to approximate the binomial distribution.
- For a binomial distribution to be approximated by a normal distribution, the conditions \( n \times p > 5 \) and \( n \times (1-p) > 5 \) must be met. This ensures the distribution is sufficiently bell-shaped.
- In our example, we set \( p = 0.675 \) (the probability of success). With \( n = 200 \), this requirement is satisfied.
- Using normal approximation, we calculate the mean \( \mu = n \times p = 135 \) and the standard deviation \( \sigma = \sqrt{n \times p \times (1-p)} \approx 6.478 \).
The Role of Standard Normal Distribution
The standard normal distribution is a key concept in probability and statistics. It is a normal distribution with a mean of 0 and a standard deviation of 1.
- By converting a normal distribution to a standard normal distribution, we can use Z-scores to easily determine probabilities and critical values.
- A Z-score measures how many standard deviations an element is from the mean. The formula is \( Z = \frac{X - \mu}{\sigma} \).
- In the given example, we converted \( X < 125 \) into a Z-score: \( Z = \frac{125 - 135}{6.478} \approx -1.542 \). This allows us to use Z-tables to find probabilities.
- Z-tables give the probability of a Z-score being below a value, here showing that the probability of fewer than 125 people being born in their state is approximately 0.0618.
Insights from U.S. Census Statistics
U.S. Census statistics provide invaluable data that forms the basis for numerous statistical analyses, including probability exercises like this one. These statistics offer insights into population characteristics and help in constructing accurate models.
- The U.S. Census data that 67.5% of Americans were born in their state of residence establishes the probability \( p = 0.675 \).
- Such statistics help scholars and researchers assume realistic probabilities, enhancing the reliability of sample data in representing whole populations.
- In our example, the sample proportion derived from a small group of 200 people is used to make inferences about the larger population 鈥� thanks to the foundational probability provided by the census data.
