Mathematics
Grade 12
15 min
Find confidence intervals for population proportions
Find confidence intervals for population proportions
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1
Introduction & Learning Objectives
Learning Objectives
Define population proportion (p) and sample proportion (p̂) and distinguish between them.
Explain how the continuous Normal distribution is used to approximate the discrete sampling distribution of a sample proportion.
State and verify the conditions necessary for constructing a valid confidence interval for a population proportion.
Calculate a confidence interval for a population proportion using the appropriate formula and critical values (z*).
Interpret a confidence interval in the context of a real-world problem.
Determine the minimum sample size required to achieve a specified margin of error for a given confidence level.
Ever wonder how a poll of just 1,500 people can claim to represent the opinion of an entire country? 🗳️ We're about to uncove...
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Key Concepts & Vocabulary
TermDefinitionExample
Population Proportion (p)The true proportion or percentage of an entire population that possesses a certain characteristic. This is a fixed, but usually unknown, value.The actual percentage of all smartphone users in Canada who use Brand X. We can't survey everyone, so 'p' is unknown.
Sample Proportion (p̂)The proportion of a sample that has a certain characteristic. It is calculated as the number of successes (x) divided by the sample size (n). It is our best point estimate for the unknown population proportion (p).In a random sample of 500 Canadian smartphone users, 150 use Brand X. The sample proportion is p̂ = 150/500 = 0.30 or 30%.
Confidence IntervalAn interval of plausible values for an unknown population parameter, such as the population propor...
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Core Formulas
One-Proportion z-Interval Formula
\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
This formula is used to calculate the confidence interval for a population proportion, p. It starts with the sample proportion (p̂) and adds/subtracts the margin of error. This is valid when the conditions for inference are met.
Conditions for Inference
1. Random: The data comes from a random sample. 2. 10% Condition: The sample size n is no more than 10% of the population size N (n ≤ 0.10N). 3. Large Counts: Both np̂ ≥ 10 and n(1-p̂) ≥ 10.
These conditions must be verified before constructing the confidence interval. The 'Large Counts' condition is crucial as it justifies using the continuous normal distribution to approximate the discrete sampling distribution of p̂.
Sample...
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Challenging
A school principal wants to compare student satisfaction at two high schools. School A's 95% CI for satisfaction is (0.72, 0.80) from a sample of 200. School B's 95% CI is (0.61, 0.69) from a sample of 250. What is the most appropriate conclusion?
A.The satisfaction levels are the same because both are above 50%.
B.There is evidence of a difference in satisfaction, as the two confidence intervals do not overlap.
C.No conclusion can be drawn because the sample sizes are different.
D.School B has a higher satisfaction rate than School A.
Challenging
A news report states: 'Our poll of 1,000 voters shows the challenger leading the incumbent 54% to 46%. The 95% confidence interval is (0.51, 0.57). This means there is a 95% chance the challenger's true support level is in this range.' What is the flaw in this interpretation?
A.The sample size is too small for this conclusion.
B.The interpretation incorrectly assigns a probability to the true proportion being in a specific, calculated interval.
C.The confidence level should be 99% for political polls.
D.The margin of error was calculated incorrectly.
Challenging
The use of a z* critical value from the continuous Normal distribution to approximate the sampling distribution of p̂ is justified by the Central Limit Theorem. Why is a 'continuity correction' factor (e.g., adding or subtracting 0.5/n) not typically used when constructing these confidence intervals in an introductory course?
A.For the large sample sizes required by the 'Large Counts' rule, the effect of the continuity correction is negligible and unnecessarily complicates the formula.
B.The continuity correction only applies when calculating probabilities, not intervals.
C.The z* critical value already has the continuity correction built into it.
D.The continuity correction was found to be statistically invalid for proportions.
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