Real-World Margin of Error Quiz (HSS.IC.B.4)

1) A poll of 1,000 adults reports that 58% support a new policy with a margin of error of ±3% (95% confidence). Which interval best represents the plausible range for true support?

55% to 61%

55% to 60%

54% to 62%

52% to 64%

58% ± 3% gives 55% to 61%.

Explanation

58% ± 3% gives 55% to 61%.

2) Two independent polls report: Poll A says 46% ± 4% support, Poll B says 51% ± 3% support (both at 95% confidence). What can you reasonably conclude?

Poll B proves support is above 50%

Poll A must be wrong because its margin of error is larger

The intervals overlap, so a real difference is not certain

The difference is statistically significant at 95%

Poll A: 42–50, Poll B: 48–54; they overlap, so no clear difference.

Explanation

Poll A: 42–50, Poll B: 48–54; they overlap, so no clear difference.

3) A sample of 400 students finds that 30% prefer online classes. Approximate the 95% margin of error for this proportion using MOE ≈ 1/√n.

± 3.2%

± 5.0%

± 1.6%

± 10.0%

1/√400 = 1/20 = 0.05 → approx 3–5%. Closest is 3.2%.

Explanation

1/√400 = 1/20 = 0.05 → approx 3–5%. Closest is 3.2%.

4) A company estimates the mean delivery time as 42 minutes with a 95% margin of error of ±4 minutes. Which is a correct interpretation?

95% of delivery times fall between 38 and 46 minutes

We are 95% confident the true mean is between 38 and 46 minutes

There is a 95% chance the sample mean is between 38 and 46 minutes

Exactly 95% of future deliveries will be between 38 and 46 minutes

Margin of error refers to the true mean.

Explanation

Margin of error refers to the true mean.

5) A media headline says “Support jumped from 48% to 50%.” Poll 1 had ±3% MOE; Poll 2 had ±2% MOE (both 95% CI). What’s the best conclusion?

The change may not be meaningful because intervals overlap

The change is definitely significant because 50% exceeds 48%

The second poll is always right because its MOE is smaller

The first poll is biased because its MOE is larger

Intervals overlap; the 2% change may be sampling variation.

Explanation

Intervals overlap; the 2% change may be sampling variation.

6) Which survey has the narrowest 95% confidence interval?

Survey 1

Survey 2

Survey 3

All have the same width

Survey 2 has the smallest margin of error.

Explanation

Survey 2 has the smallest margin of error.

7) Which pair of surveys shows the strongest evidence of a true difference in support?

Survey 1 vs Survey 2

Survey 2 vs Survey 3

Survey 1 vs Survey 3

None show any difference

All intervals overlap; no clear difference.

Explanation

All intervals overlap; no clear difference.

8) Based on the reported results, which interval is correctly stated?

Survey 1: 50% to 56%, Survey 2: 47% to 57%, Survey 3: 44% to 54%

Survey 1: 50% to 58%, Survey 2: 49.5% to 54.5%, Survey 3: 44% to 54%

Survey 1: 46% to 62%, Survey 2: 50% to 55%, Survey 3: 47% to 51%

Survey 1: 52% to 56%, Survey 2: 49% to 55%, Survey 3: 45% to 55%

Correct intervals shown.

Explanation

Correct intervals shown.

9) If the city wants the margin of error to be at most ±2% at 95% confidence for a proportion near 50%, which sample size is closest to sufficient?

2,500

625

1,000

400

0.02 ≈ 1/√n → n ≈ 2,500.

Explanation

0.02 ≈ 1/√n → n ≈ 2,500.

10) A product rating survey reports a mean score of 4.1 out of 5 with a 95% margin of error of ±0.2. Which statement is most accurate?

95% of customers rated between 3.9 and 4.3

The sample mean will change by at most 0.2 in future samples

It is plausible the true mean is between 3.9 and 4.3

The standard deviation must be 0.2

True mean likely between 3.9 and 4.3.

Explanation

True mean likely between 3.9 and 4.3.

11) A pollster doubles the sample size from 500 to 1,000. Approximately how does the 95% margin of error for a proportion change?

It doubles

It is unchanged

It halves exactly

It decreases

MOE decreases with larger sample size.

Explanation

MOE decreases with larger sample size.

12) A school reports that 62% ± 4% of parents favor a new schedule (95% CI). A competing report says 58% ± 3%. Which is the best interpretation?

The intervals overlap, so a meaningful difference is not guaranteed

The first school has higher support with certainty

The second report is wrong because its point estimate is lower

The true support must be 60%

Intervals overlap.

Explanation

Intervals overlap.

13) Which statement about the phone and online results is correct?

The online result proves vaccination is increasing

The phone survey is more accurate because it was first

The phone survey’s CI excludes the online estimate

Their confidence intervals overlap substantially

Intervals overlap substantially.

Explanation

Intervals overlap substantially.

14) What is the 95% confidence interval for the combined report?

42.7% to 49.3%

44% to 48%

45% to 47%

43.6% to 48.4%

46% ± 2.4% = 43.6% to 48.4%.

Explanation

46% ± 2.4% = 43.6% to 48.4%.

15) If the department wants to reduce the combined MOE to ±1.2% at 95% confidence (with proportion near 50%), about how many total respondents are needed?

900

6,700

4,200

1,800

Halving MOE → 4× sample → about 6,700.

Explanation

Halving MOE → 4× sample → about 6,700.

16) Which conclusion is most justified?

The online panel is biased and should be discarded

The phone survey is more reliable because it has a lower MOE

The combined estimate is more precise due to effectively larger sample size

Neither estimate is useful because they differ by 4%

Combining increases precision.

Explanation

Combining increases precision.

17) A study reports the average daily steps as 7,800 with a 90% MOE of ±400 steps. Which is a correct comparison to a 95% MOE?

The 95% MOE would be larger than ±400 steps

The 95% MOE would be smaller than ±400 steps

The 95% MOE would be exactly ±400 steps

The MOE is unrelated to confidence level

95% MOE > 90% MOE.

Explanation

95% MOE > 90% MOE.

18) A candidate’s support is 47% ± 3% (95% CI). What is the strongest conclusion?

The candidate is definitely behind

True support must be below 50%

True support could plausibly be between 44% and 50%

The sample size must be 1,000

47% ± 3% gives 44–50%.

Explanation

47% ± 3% gives 44–50%.

19) A cereal company tests a new recipe. The mean satisfaction score is 7.2/10 with MOE ±0.5 (95% CI). Another version scores 7.6/10 with MOE ±0.4 (95% CI). Which is best supported?

The new version is definitely better, no overlap

The intervals overlap, so the difference may not be statistically meaningful

The second version is worse because of smaller MOE

Both versions must be equal

Intervals overlap.

Explanation

Intervals overlap.