Math Problem Statement

Assume that a sample is used to estimate a population proportion p. Calculate the amount of variability we expect to see in our estimate (the margin of error), given a sample of size 275 with 78.2% successes at a confidence level of 99%.

Solution

To solve this problem, we need to compute the margin of error (M.E.) using the given formula for estimating a population proportion:

M.E.=zp^(1p^)n\text{M.E.} = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

Where:

  • zz^* is the critical value for a given confidence level.
  • p^\hat{p} is the sample proportion (success rate).
  • nn is the sample size.

Given Data:

  • Sample size n=275n = 275
  • Success rate p^=78.2%=0.782\hat{p} = 78.2\% = 0.782
  • Confidence level = 99% (corresponding zz^* value = 2.576)

Step 1: Compute p^(1p^)\hat{p}(1 - \hat{p})

p^(1p^)=0.782×(10.782)=0.782×0.218=0.170476\hat{p}(1 - \hat{p}) = 0.782 \times (1 - 0.782) = 0.782 \times 0.218 = 0.170476

Step 2: Plug into the formula

M.E.=2.576×0.170476275\text{M.E.} = 2.576 \times \sqrt{\frac{0.170476}{275}}

Step 3: Simplify

0.1704762750.000619\frac{0.170476}{275} \approx 0.000619 0.0006190.02488\sqrt{0.000619} \approx 0.02488 M.E.=2.576×0.024880.06406\text{M.E.} = 2.576 \times 0.02488 \approx 0.06406

Step 4: Convert to percentage

M.E.=0.06406×100=6.406%\text{M.E.} = 0.06406 \times 100 = 6.406\%

Step 5: Round to one decimal place

M.E.6.4%\text{M.E.} \approx 6.4\%

Thus, the margin of error is 6.4%.


Would you like further details on any part of the calculation?

Here are five related questions:

  1. How do we find the critical value for different confidence levels?
  2. How does increasing the sample size affect the margin of error?
  3. What is the relationship between confidence level and margin of error?
  4. Can this method be applied to other types of population parameters?
  5. How is the formula different when we don't know the population proportion?

Tip: As the confidence level increases, the critical value zz^* increases, which also increases the margin of error.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Population Proportion
Margin of Error
Confidence Interval

Formulas

Margin of Error (M.E.) = z* * sqrt(p̂(1 - p̂) / n)
p̂ = sample proportion
n = sample size
z* = critical value based on confidence level

Theorems

Central Limit Theorem
Normal Distribution

Suitable Grade Level

Grades 11-12, College