(a) What is the point estimate of the population proportion of adults who lack confidence they will be able to afford health insurance in the future. (Round your answer to two decimal places.) (b) At 90% confidence, what is the margin of error? (Round your answer to four decimal places.) (c) Develop a 90% confidence interval for the population proportion of adults who lack confidence they will be able to afford health insurance in the future. (Round your answer to four decimal places.)
(a) What is the point estimate of the population proportion of adults who lack confidence they will be able to afford health insurance in the future. (Round your answer to two decimal places.) (b) At 90% confidence, what is the margin of error? (Round your answer to four decimal places.) (c) Develop a 90% confidence interval for the population proportion of adults who lack confidence they will be able to afford health insurance in the future. (Round your answer to four decimal places.)
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 4CYU
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![### Statistical Confidence and Population Proportion
**Understanding Population Proportion and Confidence Intervals:**
In the field of statistics, confidence intervals are a crucial way to present the reliability of an estimate. A confidence interval provides an estimated range of values which is likely to include an unknown population parameter. The width of the confidence interval gives us an idea about how uncertain we are about the unknown parameter; a wider interval may suggest more uncertainty whereas a narrower interval suggests more precision.
#### Case Study: Survey on Affordability of Health Insurance
In the spring of 2017, the Consumer Reports National Research Center conducted a survey involving 1,007 adults to learn about their major healthcare concerns. The purpose of this survey was to understand their confidence in being able to afford health insurance in the future.
**Survey Result:**
- Out of 1,007 respondents, 577 adults expressed a lack of confidence in their ability to afford health insurance in the future.
Let's explore this data through a series of analytical questions:
### (a) Point Estimate of Population Proportion
**Question:**
What is the point estimate of the population proportion of adults who lack confidence they will be able to afford health insurance in the future? (Round your answer to two decimal places.)
**Answer:**
To find the point estimate (also known as the sample proportion) \( \hat{p} \):
\[ \hat{p} = \frac{\text{Number of adults lacking confidence}}{\text{Total number of respondents}} = \frac{577}{1007} \]
Perform the division to obtain the point estimate.
### (b) Margin of Error at 90% Confidence Level
**Question:**
At 90% confidence, what is the margin of error? (Round your answer to four decimal places.)
**Answer:**
To calculate margin of error (E) at a 90% confidence level, use the formula:
\[ E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Where:
- \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level.
- \( \hat{p} \) is the sample proportion.
- \( n \) is the sample size.
Look up the z-score for a 90% confidence level and plug in the values for \( \hat{p} \) and \( n \).
### (c) Confidence](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5018757-df34-47e5-8789-7c8e64b31b6d%2F99da3e1a-356e-4ed4-b191-94898b2a705d%2Fop1c57d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Statistical Confidence and Population Proportion
**Understanding Population Proportion and Confidence Intervals:**
In the field of statistics, confidence intervals are a crucial way to present the reliability of an estimate. A confidence interval provides an estimated range of values which is likely to include an unknown population parameter. The width of the confidence interval gives us an idea about how uncertain we are about the unknown parameter; a wider interval may suggest more uncertainty whereas a narrower interval suggests more precision.
#### Case Study: Survey on Affordability of Health Insurance
In the spring of 2017, the Consumer Reports National Research Center conducted a survey involving 1,007 adults to learn about their major healthcare concerns. The purpose of this survey was to understand their confidence in being able to afford health insurance in the future.
**Survey Result:**
- Out of 1,007 respondents, 577 adults expressed a lack of confidence in their ability to afford health insurance in the future.
Let's explore this data through a series of analytical questions:
### (a) Point Estimate of Population Proportion
**Question:**
What is the point estimate of the population proportion of adults who lack confidence they will be able to afford health insurance in the future? (Round your answer to two decimal places.)
**Answer:**
To find the point estimate (also known as the sample proportion) \( \hat{p} \):
\[ \hat{p} = \frac{\text{Number of adults lacking confidence}}{\text{Total number of respondents}} = \frac{577}{1007} \]
Perform the division to obtain the point estimate.
### (b) Margin of Error at 90% Confidence Level
**Question:**
At 90% confidence, what is the margin of error? (Round your answer to four decimal places.)
**Answer:**
To calculate margin of error (E) at a 90% confidence level, use the formula:
\[ E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Where:
- \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level.
- \( \hat{p} \) is the sample proportion.
- \( n \) is the sample size.
Look up the z-score for a 90% confidence level and plug in the values for \( \hat{p} \) and \( n \).
### (c) Confidence
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