Confidence Interval for a Population Proportion
Construct a 95% confidence interval for a population proportion using sample data from a survey about smartphone usage.
Problem Scenario
A market research firm surveys 400 randomly selected adults and finds that 272 of them use a smartphone for more than 3 hours per day. Construct a 95% confidence interval for the true proportion of all adults who use their smartphone for more than 3 hours daily.
Given Data
Requirements
- Calculate the sample proportion and verify the conditions for the normal approximation
- Construct the 95% confidence interval for the population proportion
- Interpret the interval in context
Solution
Step 1:
Calculate the sample proportion: p-hat = x / n = 272 / 400 = 0.68.
Step 2:
Check conditions: np-hat = 400(0.68) = 272 >= 10 and n(1 - p-hat) = 400(0.32) = 128 >= 10. Both conditions are satisfied, so the normal approximation is valid.
Step 3:
Calculate the standard error: SE = sqrt[p-hat(1 - p-hat) / n] = sqrt[(0.68)(0.32) / 400] = sqrt[0.2176 / 400] = sqrt(0.000544) = 0.02332.
Step 4:
Calculate the margin of error: ME = z* x SE = 1.96 x 0.02332 = 0.04571.
Step 5:
Construct the interval: p-hat +/- ME = 0.68 +/- 0.04571, giving (0.6343, 0.7257). Rounding appropriately: (0.634, 0.726).
Final Answer
The 95% confidence interval for the population proportion is (0.634, 0.726). We are 95% confident that between 63.4% and 72.6% of all adults use their smartphone for more than 3 hours per day.
Key Takeaways
- ✓The width of a confidence interval decreases as sample size increases, because a larger sample gives more precise estimates.
- ✓A 95% confidence level means that if we repeated this sampling procedure many times, about 95% of the resulting intervals would contain the true population proportion.
- ✓Always check the success/failure condition (np-hat >= 10 and n(1 - p-hat) >= 10) before using the normal approximation.
Common Errors to Avoid
- ✗Interpreting the interval as "there is a 95% probability the true proportion falls in this interval." The true proportion is a fixed value; the confidence refers to the procedure, not this specific interval.
- ✗Forgetting to check the normality conditions before applying the formula.
- ✗Using the wrong z* critical value. For 95% confidence, z* = 1.96; for 99%, z* = 2.576; for 90%, z* = 1.645.
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Common questions about this problem type
Higher confidence levels produce wider intervals. A 95% level is most common and offers a good balance between precision and confidence. Use 99% when the consequences of being wrong are severe (e.g., medical studies). Use 90% when you can tolerate more risk and want a narrower interval.
If np-hat < 10 or n(1 - p-hat) < 10, use an exact method such as the Clopper-Pearson interval or the Wilson score interval instead of the Wald (normal approximation) interval.