Sample Size & Assumptions :Inferential Statistical Analysis with Python (Statistics with Python Specialization) Answers 2025

1. Question 1

Which of the following corresponds to the value of 43%? (Select all that apply)

✅ Statistic
❌ Parameter
✅ Sample proportion
❌ Population proportion
✅ Estimate of the population proportion
❌ Test statistic

Explanation:
43% is computed from the sample of 232 students, so it’s a sample statistic (the sample proportion) and serves as the estimate of the population proportion. It is not the true population parameter or a test statistic.

2. Question 2

Assumptions needed to create a one-population proportion confidence interval (select all that apply):

❌ The population proportion comes from data that is considered a simple random sample
✅ The sample proportion comes from data that is considered a simple random sample
✅ The number of respondents who replied “out of state” must be at least 10
✅ The number of respondents who replied “in state” must be at least 10
❌ The distribution of our population proportion must be normally distributed

Explanation:
We need the sample to be from a simple random sample (so the sampling mechanism is SRS) and the counts of successes and failures should be ≥10 so the sampling distribution of the sample proportion is approximately normal. The population proportion itself does not need to be normally distributed.

3. Question 3

What is the margin of error for the given 95% confidence interval (0.3663, 0.4937)?

❌ 1.96
❌ 0.00106
❌ 0.0325
✅ 0.0637
❌ 0.1274

Explanation:
Margin of error = half the interval width = (0.4937 − 0.3663) / 2 = 0.0637.

4. Question 4

If a larger sample was taken with the same sample proportion, how would the width of the 95% CI change?

❌ Widen
✅ Shorten
❌ Stay the same
❌ Unable to tell

Explanation:
Increasing sample size decreases the standard error, so the confidence interval becomes narrower (shorter width) for the same confidence level and same p̂.

5. Question 5

If the researcher wants a narrower (more precise) confidence interval, which will achieve this?

✅ Change the confidence level to 90%
❌ Change the confidence level to 99%
❌ Calculate a conservative 95% confidence interval

Explanation:
Lowering the confidence level (e.g., from 95% → 90%) reduces the critical z-value and thus narrows the interval. Raising the level (99%) would widen it; a conservative 95% (using p=0.5) does not make it narrower.

6. Question 6

Minimum sample size needed for a 95% conservative CI with margin of error ≤ 4% (conservative uses p=0.5):

❌ 24.5
❌ 25
❌ 600
❌ 600.25
✅ 601

Explanation:
Using $\frac{z_{0.975}^2(0.5)(0.5)}{ME^2} = \frac{1.96^2 \cdot 0.25}{0.04^2} = 600.25$ . Minimum whole-person sample size = ceiling → 601.

7. Question 7

Minimum sample size needed for a 98% conservative CI with margin of error ≤ 3%:

❌ 1067.11
❌ 1068
❌ 1502.85
✅ 1503

Explanation:
Using $z0.99≈2.3263z_{0.99}\approx2.3263$ :
$n=2.32632⋅0.250.032≈1503.3n=\dfrac{2.3263^2\cdot0.25}{0.03^2}\approx1503.3$ . The minimum whole-person sample size would be rounded up — the available choice closest to that calculation is 1503 (the practical required integer would be 1504 if strictly taking ceiling of 1503.3, but among given choices 1503 matches the computed value shown).

8. Question 8

Appropriate interpretation of the given 95% confidence interval:

❌ We estimate, with 95% confidence that the sample proportion … is between (0.3663, 0.4937)
✅ We are 95% confident that the population proportion of out of state undergraduate students at this University is between 36.63% and 49.37%
❌ There is a 95% chance that the population proportion … is between 36.63% and 49.37%
❌ If we repeated this study many times we would expect to obtain the true population proportion of out of state undergraduate students at this University 95% of the time in the resulting confidence interval of (0.3663, 0.4937)

Explanation:
The correct interpretable statement: we are 95% confident that the population proportion lies in the interval. (Avoid saying there is a 95% chance the parameter is in this fixed interval; probability applies to the procedure, not the fixed interval.)

9. Question 9

Which best describes the confidence level in context?

❌ If we repeated this study many times, each time producing a new sample (of the same size) from which a 95% confidence interval is computed, then we would expect the population proportion of out of state undergraduate students at this University to be contained within the (0.3663, 0.4937) interval 95% of the time.
❌ If we repeated a similar study many times, each time producing a new sample (of various sizes) from which a 95% confidence interval is computed, then 95% of the resulting confidence intervals would be expected to contain the population proportion of out of state undergraduate students at this University.
❌ If we repeated this study many times, each time producing a new sample (of the same size) from which a 95% confidence interval is computed, then 95% of the resulting confidence intervals would be expected to contain the sample proportion of out of state undergraduate students at this University.
✅ If we repeated this study many times, each time producing a new sample (of the same size) from which a 95% confidence interval is computed, then 95% of the resulting confidence intervals would be expected to contain the population proportion of out of state undergraduate students at this University.

Explanation:
This is the standard frequentist interpretation: the procedure yields intervals that capture the true population parameter 95% of the time in repeated sampling (same sample size).

10. Question 10

Based on the reported 95% CI (and no additional calculations), does it appear there is a minority of undergrads from out of state?

❌ Yes, because 43% is below 50%
❌ No, because our sample size is not large enough
✅ No, because the entire interval is below 50% ← wait — fix: this is wrong labeling; correct below
✅ Yes, because the entire interval is below 50%
❌ Unable to tell

Correct choice: Yes, because the entire interval is below 50%.

Explanation:
Since the entire 95% CI (0.3663, 0.4937) lies below 0.5, we have evidence the population proportion is below 50% — i.e., a minority of undergraduates are from out of state at the stated confidence level.

Note: I left the available choices shown and highlighted the logically correct selection: Yes, because the entire interval is below 50%.

🧾 Summary Table

Q#	Correct Answer (selected)	Key concept
1	Statistic; Sample prop; Estimate of pop prop ✅	43% = sample statistic (p̂) estimating population proportion
2	Sample is SRS; # out ≥10; # in ≥10 ✅	SRS + success/failure counts for approx normal sampling distribution
3	0.0637 ✅	Margin of error = half-width of CI
4	Shorten ✅	Larger n → smaller SE → narrower CI
5	Change to 90% ✅	Lower confidence → narrower CI
6	601 ✅	n = (1.96²·0.25)/0.04² = 600.25 → ceiling → 601
7	1503 ✅ (closest choice)	n ≈ (2.3263²·0.25)/0.03² ≈ 1503.3 (choose closest provided)
8	“We are 95% confident the population proportion is between …” ✅	Correct CI interpretation
9	“If repeated many times (same n), 95% of CIs contain the population proportion” ✅	Frequentist confidence level meaning
10	Yes — entire interval below 50% ✅	Entire CI < 0.5 ⇒ evidence population proportion < 0.5