The population proportion is 0.20. What is the probability that a sample proportion will be within 10.04 of the population proportion for each of the following sample sizes? (Round your answers to 4 decimal places.) (a) n = 100 (b) n = 200 (c) n = 500 (d) n = 1,000 X (e) What is the advantage of a larger sample size? O There is a higher probability will be within 10.04 of the population standard deviation. O There is a higher probability p will be within +0.04 of the population proportion p. O We can guarantee p will be within ±0.04 of the population proportion p. O As sample size increases, E(p) approaches p.
The population proportion is 0.20. What is the probability that a sample proportion will be within 10.04 of the population proportion for each of the following sample sizes? (Round your answers to 4 decimal places.) (a) n = 100 (b) n = 200 (c) n = 500 (d) n = 1,000 X (e) What is the advantage of a larger sample size? O There is a higher probability will be within 10.04 of the population standard deviation. O There is a higher probability p will be within +0.04 of the population proportion p. O We can guarantee p will be within ±0.04 of the population proportion p. O As sample size increases, E(p) approaches p.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Understanding Sampling Probability in Population Proportions
The population proportion is given as \( p = 0.20 \). The task is to determine the probability that a sample proportion will fall within ±0.04 of the population proportion for the given sample sizes. The answers should be rounded to four decimal places.
**Sample Sizes and Probability Computations:**
(a) \( n = 100 \)
- [Answer to be computed]
(b) \( n = 200 \)
- [Answer to be computed]
(c) \( n = 500 \)
- [Answer to be computed]
(d) \( n = 1,000 \)
- [Answer to be computed]
**Interpreting Larger Sample Sizes:**
(e) What is the advantage of a larger sample size?
Possible options are:
1. There is a higher probability \(\sigma_{\bar{p}}\) will be within ±0.04 of the population standard deviation.
2. There is a higher probability \(\bar{p}\) will be within ±0.04 of the population proportion \( p \).
3. We can guarantee \(\bar{p}\) will be within ±0.04 of the population proportion \( p \).
4. As sample size increases, \( E(\bar{p}) \) approaches \( p \).
### Detailed Explanation
**Understanding Graphs and Diagrams:**
This section contains text boxes and radio buttons for inputting sample probabilities and selecting the correct statements. There are no visual graphs or complex diagrams, but the information looks to be organized in a Q&A format for educational purposes.
**Instruction on Calculating Probabilities:**
To solve for each part (a-d), one would typically standardize the sampling distribution of the sample portion \(\hat{p}\) using:
\[ \hat{p} \text{ is approximately } N(p, \sqrt{\frac{p(1-p)}{n}}) \]
- Calculate the standard error (SE) for each sample size \( n \).
Using the Standard Normal Distribution (Z-distribution), center it around
\[ Z = \frac{\hat{p} - p}{\sqrt{\frac{p (1 - p)}{n}}} \]
By applying this standard, you can determine the values to plug in to calculate each probability and answer the sample size advantage question.
This framework helps in understanding how sample size influences the precision of estimates in statistical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F928f3826-faea-4936-b5b6-4aab28469498%2F1fae955a-c72c-459e-81b5-875026a7f96f%2Fqsmdgbr_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Sampling Probability in Population Proportions
The population proportion is given as \( p = 0.20 \). The task is to determine the probability that a sample proportion will fall within ±0.04 of the population proportion for the given sample sizes. The answers should be rounded to four decimal places.
**Sample Sizes and Probability Computations:**
(a) \( n = 100 \)
- [Answer to be computed]
(b) \( n = 200 \)
- [Answer to be computed]
(c) \( n = 500 \)
- [Answer to be computed]
(d) \( n = 1,000 \)
- [Answer to be computed]
**Interpreting Larger Sample Sizes:**
(e) What is the advantage of a larger sample size?
Possible options are:
1. There is a higher probability \(\sigma_{\bar{p}}\) will be within ±0.04 of the population standard deviation.
2. There is a higher probability \(\bar{p}\) will be within ±0.04 of the population proportion \( p \).
3. We can guarantee \(\bar{p}\) will be within ±0.04 of the population proportion \( p \).
4. As sample size increases, \( E(\bar{p}) \) approaches \( p \).
### Detailed Explanation
**Understanding Graphs and Diagrams:**
This section contains text boxes and radio buttons for inputting sample probabilities and selecting the correct statements. There are no visual graphs or complex diagrams, but the information looks to be organized in a Q&A format for educational purposes.
**Instruction on Calculating Probabilities:**
To solve for each part (a-d), one would typically standardize the sampling distribution of the sample portion \(\hat{p}\) using:
\[ \hat{p} \text{ is approximately } N(p, \sqrt{\frac{p(1-p)}{n}}) \]
- Calculate the standard error (SE) for each sample size \( n \).
Using the Standard Normal Distribution (Z-distribution), center it around
\[ Z = \frac{\hat{p} - p}{\sqrt{\frac{p (1 - p)}{n}}} \]
By applying this standard, you can determine the values to plug in to calculate each probability and answer the sample size advantage question.
This framework helps in understanding how sample size influences the precision of estimates in statistical analysis.
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