Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 163 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that fewer than 42 voted The probability that fewer than 42 of 163 eligible voters voted is (Round to four decimal places as needed.)
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 163 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that fewer than 42 voted The probability that fewer than 42 of 163 eligible voters voted is (Round to four decimal places as needed.)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![**Finding the Probability Using Normal Approximation**
To find the probability of the indicated number of voters, we will use a normal approximation. In this case, assume that 163 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted.
**Problem Statement:**
Find the probability that fewer than 42 voters voted.
**Given Data:**
- Total eligible voters (n): 163
- Proportion of voters (p): 22% (or 0.22)
**Required Probability:**
The probability that fewer than 42 of 163 eligible voters voted.
**Solution Outline:**
1. Calculate the mean (μ) and standard deviation (σ) of the binomial distribution.
Formula for mean (μ):
\[
\mu = np
\]
Formula for standard deviation (σ):
\[
\sigma = \sqrt{np(1-p)}
\]
2. Convert the binomial probability to a normal probability by using the normal approximation to the binomial distribution.
3. Find the z-score for 42 voters.
4. Use the z-score to find the corresponding probability from the standard normal distribution table.
**Note:**
- Remember to round your final answer to four decimal places as needed.
---
**Note to the students:**
This example illustrates how to use the normal approximation to compute probabilities for a binomial distribution, especially when dealing with large sample sizes.
For further explanation and detailed step-by-step solutions, refer to the section on "Normal Approximation to the Binomial Distribution" in your statistics textbook.
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Transcribed Image Text:**Finding the Probability Using Normal Approximation**
To find the probability of the indicated number of voters, we will use a normal approximation. In this case, assume that 163 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted.
**Problem Statement:**
Find the probability that fewer than 42 voters voted.
**Given Data:**
- Total eligible voters (n): 163
- Proportion of voters (p): 22% (or 0.22)
**Required Probability:**
The probability that fewer than 42 of 163 eligible voters voted.
**Solution Outline:**
1. Calculate the mean (μ) and standard deviation (σ) of the binomial distribution.
Formula for mean (μ):
\[
\mu = np
\]
Formula for standard deviation (σ):
\[
\sigma = \sqrt{np(1-p)}
\]
2. Convert the binomial probability to a normal probability by using the normal approximation to the binomial distribution.
3. Find the z-score for 42 voters.
4. Use the z-score to find the corresponding probability from the standard normal distribution table.
**Note:**
- Remember to round your final answer to four decimal places as needed.
---
**Note to the students:**
This example illustrates how to use the normal approximation to compute probabilities for a binomial distribution, especially when dealing with large sample sizes.
For further explanation and detailed step-by-step solutions, refer to the section on "Normal Approximation to the Binomial Distribution" in your statistics textbook.
(End of transcription)
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