Question 1178708: Suppose in 2011 a survey of 946 workers in the city of Montreal found that 214 used public transport to commute to work.
Part (a) Estimate the percentage of workers in Montreal who use public transport to commute to work. Give your answer in percentages (without the percentage sign, and to four significant digits). .
Part (b) Find a 95% confidence interval for the percentage of workers in Montreal using public transport to commute to work. Use the sample proportion in your estimate of the standard error. Provide the upper and lower bounds for your confidence interval. Give your answer in percentages (without the percentage sign, and to four significant digits) ( , )
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step:
**Part (a) Estimate the Percentage:**
1. **Calculate the Sample Proportion (p̂):**
* p̂ = (Number of workers using public transport) / (Total number of workers)
* p̂ = 214 / 946
* p̂ ≈ 0.2262156448
2. **Convert to Percentage:**
* Percentage = p̂ * 100%
* Percentage ≈ 0.2262156448 * 100%
* Percentage ≈ 22.62156448%
3. **Round to Four Significant Digits:**
* Percentage ≈ 22.62%
**Answer:** 22.62
**Part (b) 95% Confidence Interval:**
1. **Calculate the Standard Error (SE):**
* SE = √[p̂(1 - p̂) / n]
* SE = √[0.2262(1 - 0.2262) / 946]
* SE = √[0.2262(0.7738) / 946]
* SE = √[0.17499956 / 946]
* SE = √0.00018498896
* SE ≈ 0.0136
2. **Find the Z-score for a 95% Confidence Interval:**
* For a 95% confidence interval, the z-score is 1.96.
3. **Calculate the Margin of Error (ME):**
* ME = z * SE
* ME = 1.96 * 0.0136
* ME ≈ 0.026656
4. **Calculate the Confidence Interval:**
* Lower Bound: p̂ - ME = 0.2262 - 0.026656 ≈ 0.199544
* Upper Bound: p̂ + ME = 0.2262 + 0.026656 ≈ 0.252856
5. **Convert to Percentages:**
* Lower Bound: 0.199544 * 100% ≈ 19.9544%
* Upper Bound: 0.252856 * 100% ≈ 25.2856%
6. **Round to Four Significant Digits:**
* Lower Bound: 19.95%
* Upper Bound: 25.29%
**Answer:** (19.95, 25.29)
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