Sampling Variability & MOE Quiz (HSS.IC.B.4)

1) Using the given margin of error rule, what is the 95% margin of error for the estimate?

0.070

0.116

0.035

0.0175

Margin of error ≈ 2 × 0.035 = 0.070.

Explanation

Margin of error ≈ 2 × 0.035 = 0.070.

2) Sampling variability is:

Natural fluctuation in random samples

Difference between mean & median

Occurs only in biased samples

Disappears when n < 30

Sampling variability = natural differences across random samples.

Explanation

Sampling variability = natural differences across random samples.

3) A school estimates mean books read: 2.8 with CI (2.4, 3.2). Which is correct?

95% of students read 2.4–3.2

Method captures true mean 95% of time

True mean = 2.8

Sample mean always in interval

The 95% CI refers to long-run capture rate.

Explanation

The 95% CI refers to long-run capture rate.

4) A survey of 600 adults finds 54% support with ±4% MOE. Interpretation?

True support is between 50% and 58%

95% support the plan

MOE covers all error

54% of all adults support the plan

54% ± 4% → CI of 50% to 58%.

Explanation

54% ± 4% → CI of 50% to 58%.

5) If the city wants to cut the margin of error in half, what should they do?

Increase sample size by 2

Increase sample size by 4

Decrease sample size by 2

Increase sample size by 8

MOE is halved by multiplying the sample size by 4.

Explanation

MOE is halved by multiplying the sample size by 4.

6) What is the best point estimate for the population mean daily screen time?

3.9 hours

4.3 hours

3.5 hours

0.4 hours

The best estimate is the mean of the sample means, 3.9 hours.

Explanation

The best estimate is the mean of the sample means, 3.9 hours.

7) If the organization doubled the sample size to n=400, what is the most likely impact on the margin of error?

It would decrease

It would stay the same

It would double

It would increase slightly

A larger sample size usually makes the margin of error smaller.

Explanation

A larger sample size usually makes the margin of error smaller.

8) What is the approximate 95% margin of error for the mean?

0.4 hours

1.6 hours

0.2 hours

0.8 hours

Margin of error ≈ 2 × 0.4 = 0.8 hours.

Explanation

Margin of error ≈ 2 × 0.4 = 0.8 hours.

9) Which 95% confidence interval correctly reflects the estimate?

(3.5, 4.3)

(3.9, 4.7)

(3.1, 4.7)

(3.0, 4.8)

Mean ± margin = 3.9 ± 0.4 → (3.5, 4.3).

Explanation

Mean ± margin = 3.9 ± 0.4 → (3.5, 4.3).

10) If another study reports CI (3.6, 4.2), how does it compare?

Do not overlap; inconsistent

Both incorrect

Do not overlap; sample sizes differ

Overlap; consistent with same population

The intervals overlap, so both estimates are consistent.

Explanation

The intervals overlap, so both estimates are consistent.

11) If a new single sample of 200 students yields 0.62 preferring the library, which statement best explains this result?

It proves the population proportion increased to 0.62

It shows the simulation mean 0.58 was wrong

It indicates the sample size was too small to be useful

It is a plausible sample outcome due to sampling variability

0.62 is close to 0.58 and within the margin of error.

Explanation

0.62 is close to 0.58 and within the margin of error.

12) Two candidates: X: 48% ±3%; Y: 51% ±3%. Conclusion?

Y will definitely win

X leads

Race too close to call

Invalid results

Intervals overlap; race is statistically too close to call.

Explanation

Intervals overlap; race is statistically too close to call.

13) Which 95% confidence interval corresponds to the estimate from the simulations?

(0.51, 0.65)

(0.55, 0.61)

(0.58, 0.65)

(0.49, 0.67)

Interval = 0.58 ± 0.07 → (0.51, 0.65).

Explanation

Interval = 0.58 ± 0.07 → (0.51, 0.65).

14) A club reports 40% ± 6%. How to reduce sampling variability?

Ask more detailed questions

Survey only active members

Increase random sample size

Use voluntary poll

Larger random samples reduce sampling variability.

Explanation

Larger random samples reduce sampling variability.

15) Desired MOE ±2% at 95%: sample size?

5,000

2,500

500

250

Using MOE ≈ 1/√n → n ≈ 2500.

Explanation

Using MOE ≈ 1/√n → n ≈ 2500.

16) Sample n=900 → p̂=0.46, MOE ±3.3%. Smaller MOE?

Use nonrandom sample

Smaller sample

Sample until p̂ ≈ 0.50

Larger sample

Larger sample reduces MOE.

Explanation

Larger sample reduces MOE.

17) Which of the following is the best point estimate of the population proportion who prefer the library based on the simulations?

0.62

0.50

0.035

0.58

The best point estimate is the mean of the sample proportions, which is 0.58.

Explanation

The best point estimate is the mean of the sample proportions, which is 0.58.

18) Poll: 52% ±3%. Which is NOT justified?

Sample of 1,000

CI (49%, 55%)

Exactly 95% support

Reflects random sampling uncertainty

95% CI does not mean 95% support.

Explanation

95% CI does not mean 95% support.

19) Poll A (n=400): 60% ±5%; Poll B (n=1600): 58% ±2.5%. Explanation?

Larger samples → smaller MOE

Poll A biased

Poll B oversampled

Poll A far from 0.5

B’s larger sample size explains smaller MOE.

Explanation

B’s larger sample size explains smaller MOE.

20) Simulating samples of n=50 from population mean 70, SD 10:

Uniform 60–80

Match pop shape

Approx normal centered at 70

SD always 10

CLT: sample means ~ normal centered at true mean.

Explanation

CLT: sample means ~ normal centered at true mean.

Interpreting Sampling Variability and Margin of Error Quiz

1) Using the given margin of error rule, what is the 95% margin of error for the estimate?

2)

What first name or nickname would you like us to use?

2) Sampling variability is:

3) A school estimates mean books read: 2.8 with CI (2.4, 3.2). Which is correct?

4) A survey of 600 adults finds 54% support with ±4% MOE. Interpretation?

5) If the city wants to cut the margin of error in half, what should they do?

6) What is the best point estimate for the population mean daily screen time?

7) If the organization doubled the sample size to n=400, what is the most likely impact on the margin of error?

8) What is the approximate 95% margin of error for the mean?

9) Which 95% confidence interval correctly reflects the estimate?

10) If another study reports CI (3.6, 4.2), how does it compare?

11) If a new single sample of 200 students yields 0.62 preferring the library, which statement best explains this result?

12) Two candidates: X: 48% ±3%; Y: 51% ±3%. Conclusion?

13) Which 95% confidence interval corresponds to the estimate from the simulations?

14) A club reports 40% ± 6%. How to reduce sampling variability?

15) Desired MOE ±2% at 95%: sample size?

16) Sample n=900 → p̂=0.46, MOE ±3.3%. Smaller MOE?

17) Which of the following is the best point estimate of the population proportion who prefer the library based on the simulations?

18) Poll: 52% ±3%. Which is NOT justified?

19) Poll A (n=400): 60% ±5%; Poll B (n=1600): 58% ±2.5%. Explanation?

20) Simulating samples of n=50 from population mean 70, SD 10: