A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 71 of the 727 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible. a. With 95% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between and b. If many groups of 727 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about percent will not contain the true population proportion.
A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 71 of the 727 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible. a. With 95% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between and b. If many groups of 727 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about percent will not contain the true population proportion.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![**Confidence Interval and Population Proportion Analysis**
A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain. 71 of the 727 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.
**Questions:**
a. With 95% confidence, the proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain is between
[ ] and [ ].
b. If many groups of 727 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About [ ] percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain and about [ ] percent will not contain the true population proportion.
**Hint:**
- **Hints**: [ ]
- **Video**: [+]
**Explanation:**
Understanding confidence intervals and population proportions is essential in statistical analysis. A confidence interval provides a range of values which is likely to contain the population parameter with a certain degree of confidence. When constructing a 95% confidence interval, it means that if we were to take numerous samples and construct a confidence interval from each sample, approximately 95% of these intervals will contain the true population proportion, while 5% will not.
**Steps to Calculate a Confidence Interval for Proportion:**
1. **Calculate the sample proportion (p̂):**
\[
p̂ = \frac{\text{number of successes}}{\text{sample size}}
\]
Here, number of successes = 71 and sample size = 727.
2. **Calculate the standard error (SE) of the sample proportion:**
\[
SE = \sqrt{\frac{p̂ (1 - p̂)}{n}}
\]
Where \( p̂ \) is the sample proportion and \( n \) is the sample size.
3. **Find the critical value (z*) for a 95% confidence level:**
The critical value for a 95% confidence level is 1.96.
4. **Construct the confidence interval:**
\[
p̂ \pm (z](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53c2050e-df40-4c0a-8720-81a391ef4121%2F313ea11a-d8c7-4622-82b1-fe901bfe66c1%2Fewrvgj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Confidence Interval and Population Proportion Analysis**
A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain. 71 of the 727 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.
**Questions:**
a. With 95% confidence, the proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain is between
[ ] and [ ].
b. If many groups of 727 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About [ ] percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain and about [ ] percent will not contain the true population proportion.
**Hint:**
- **Hints**: [ ]
- **Video**: [+]
**Explanation:**
Understanding confidence intervals and population proportions is essential in statistical analysis. A confidence interval provides a range of values which is likely to contain the population parameter with a certain degree of confidence. When constructing a 95% confidence interval, it means that if we were to take numerous samples and construct a confidence interval from each sample, approximately 95% of these intervals will contain the true population proportion, while 5% will not.
**Steps to Calculate a Confidence Interval for Proportion:**
1. **Calculate the sample proportion (p̂):**
\[
p̂ = \frac{\text{number of successes}}{\text{sample size}}
\]
Here, number of successes = 71 and sample size = 727.
2. **Calculate the standard error (SE) of the sample proportion:**
\[
SE = \sqrt{\frac{p̂ (1 - p̂)}{n}}
\]
Where \( p̂ \) is the sample proportion and \( n \) is the sample size.
3. **Find the critical value (z*) for a 95% confidence level:**
The critical value for a 95% confidence level is 1.96.
4. **Construct the confidence interval:**
\[
p̂ \pm (z
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