Math Problem Statement
You are looking at a population and are interested in the proportion p that has a certain characteristic. Unknown to you, this population proportion is p=0.55 . You have taken a random sample of size n=125 from the population and found that the proportion of the sample that has the characteristic is sample proportion=0.49. Your sample is Sample 1 in the table. Sample 1, graph the 75% and 90% confidence intervals for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.645 for the critical value for the 90% confidence interval.
Solution
To calculate the confidence intervals for the population proportion based on the sample proportion, we will use the formula for a confidence interval (CI) for a population proportion:
Where:
- is the sample proportion (in this case, ),
- is the critical value (which changes depending on the confidence level),
- is the sample size ().
Step 1: 75% Confidence Interval
For the 75% confidence interval, the critical value .
The formula becomes:
Let's compute this.
Step 2: 90% Confidence Interval
For the 90% confidence interval, the critical value .
The formula becomes:
Let me calculate both of these confidence intervals.The confidence intervals for the population proportion based on the sample are as follows:
-
75% Confidence Interval:
- Lower bound: 0.439
- Upper bound: 0.541
-
90% Confidence Interval:
- Lower bound: 0.416
- Upper bound: 0.564
These intervals provide a range of values within which the true population proportion is likely to lie with the given levels of confidence.
Would you like to see these intervals plotted on a graph or need further details?
Here are 5 related questions to consider:
- How do confidence intervals change with larger sample sizes?
- What is the effect of a higher confidence level on the width of the interval?
- How is the critical value for a confidence interval determined?
- What are the assumptions behind using a normal approximation for confidence intervals of proportions?
- How does the standard error of the sample proportion change with different values of ?
Tip: Higher confidence levels result in wider intervals to ensure greater certainty that the population parameter lies within the range.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Proportion Estimation
Formulas
CI = p̂ ± z * sqrt(p̂(1 - p̂) / n)
Theorems
Central Limit Theorem
Normal Approximation for Proportions
Suitable Grade Level
Grades 11-12
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