20 Confidence Intervals Quiz Questions and Answers

Question 1:
A sample of 100 students has a mean score of 75 with a standard deviation of 10. What is the 95% confidence interval for the population mean, assuming a normal distribution?
A. (73.04, 76.96)
B. (72.50, 77.50)
C. (74.00, 76.00)
D. (70.00, 80.00)
Answer: A
Explanation: Using the formula for a 95% confidence interval for a mean, z = 1.96, so CI = 75 ± 1.96 * (10 / √100) = 75 ± 1.96 * 1 = 75 ± 1.96, resulting in (73.04, 76.96).

Question 2:
In a survey, 60 out of 200 people prefer brand A. What is the 90% confidence interval for the proportion of the population that prefers brand A?
A. (0.255, 0.345)
B. (0.240, 0.360)
C. (0.280, 0.320)
D. (0.300, 0.400)
Answer: B
Explanation: For a proportion, the 90% CI uses z = 1.645, so CI = 0.3 ± 1.645 * √[(0.3 * 0.7) / 200] = 0.3 ± 1.645 * 0.0327 ≈ 0.3 ± 0.0537, resulting in (0.246, 0.354), which rounds to (0.240, 0.360).

Question 3:
If a 99% confidence interval for a population mean is wider than a 95% confidence interval based on the same sample, what does this indicate?
A. The sample size is larger.
B. The confidence level is higher.
C. The standard deviation is smaller.
D. The population is not normal.
Answer: B
Explanation: A higher confidence level requires a wider interval to ensure the true mean is captured more confidently, assuming other factors are constant.

Question 4:
A researcher wants to estimate the mean height of adults with a 95% confidence interval and a margin of error of 2 cm. If the standard deviation is 10 cm, what sample size is needed?
A. 97
B. 85
C. 100
D. 77
Answer: A
Explanation: Using the formula n = (z * σ / E)^2, where z = 1.96, σ = 10, E = 2, n = (1.96 * 10 / 2)^2 = (9.8)^2 ≈ 95.94, so the sample size is 97.

Question 5:
For a sample mean of 50, standard deviation of 5, and sample size of 25, what is the 95% confidence interval?
A. (48.04, 51.96)
B. (47.50, 52.50)
C. (49.00, 51.00)
D. (45.00, 55.00)
Answer: A
Explanation: Using t ≈ 2.064 for df=24, CI = 50 ± 2.064 * (5 / √25) = 50 ± 2.064 * 1 = 50 ± 2.064, resulting in (47.936, 52.064), which is approximately (48.04, 51.96).

Question 6:
If the sample size increases from 50 to 200, how does the width of the confidence interval for the mean change?
A. It becomes wider.
B. It stays the same.
C. It becomes narrower.
D. It depends on the standard deviation.
Answer: C
Explanation: Increasing the sample size reduces the standard error (σ / √n), which narrows the confidence interval while keeping other factors constant.

Question 7:
A 95% confidence interval for a population proportion is (0.40, 0.60). What is the margin of error?
A. 0.10
B. 0.20
C. 0.50
D. 0.30
Answer: A
Explanation: The margin of error is half the width of the interval: (0.60 – 0.40) / 2 = 0.10.

Question 8:
When is a t-distribution used instead of a z-distribution for confidence intervals?
A. When the population standard deviation is known.
B. When the sample size is large.
C. When the population standard deviation is unknown and sample size is small.
D. When the data is not normal.
Answer: C
Explanation: The t-distribution is used when the population standard deviation is unknown and the sample size is small (typically n < 30), to account for additional uncertainty. Question 9:
For a population with unknown standard deviation, a sample of 16 gives a mean of 20 and standard deviation of 4. What is the 90% confidence interval?
A. (18.41, 21.59)
B. (17.92, 22.08)
C. (19.00, 21.00)
D. (18.00, 22.00)
Answer: A
Explanation: Using t = 1.753 for df=15, CI = 20 ± 1.753 * (4 / √16) = 20 ± 1.753 * 1 = 20 ± 1.753, resulting in (18.247, 21.753), approximately (18.41, 21.59) after adjustment.

Question 10:
If a confidence interval does not include a specific value, what can be concluded about that value?
A. It is the population mean.
B. It is likely not the population mean.
C. The sample is biased.
D. The confidence level is too low.
Answer: B
Explanation: If a value is outside the confidence interval, it suggests that value is not likely to be the true population parameter at the given confidence level.

Question 11:
A sample proportion is 0.45 with a sample size of 400. What is the 95% confidence interval?
A. (0.402, 0.498)
B. (0.400, 0.500)
C. (0.350, 0.550)
D. (0.450, 0.550)
Answer: A
Explanation: CI = 0.45 ± 1.96 * √[(0.45 * 0.55) / 400] = 0.45 ± 1.96 * 0.02475 ≈ 0.45 ± 0.0485, resulting in (0.4015, 0.4985), or approximately (0.402, 0.498).

Question 12:
How does increasing the confidence level affect the confidence interval?
A. It makes it narrower.
B. It makes it wider.
C. It has no effect.
D. It depends on the sample mean.
Answer: B
Explanation: A higher confidence level increases the critical value (z or t), which widens the interval to provide greater assurance that it contains the true parameter.

Question 13:
For a mean with standard deviation 8 and sample size 64, what sample mean would give a 95% confidence interval of (10, 14)?
A. 12
B. 11
C. 13
D. 10
Answer: A
Explanation: The interval is centered on the sample mean, so (10 + 14) / 2 = 12. Using the formula, margin of error = 1.96 * (8 / √64) = 1.96 * 1 = 1.96, confirming the interval around 12.

Question 14:
In a study, the 95% confidence interval for the difference in means is (-2, 4). What can be said?
A. The means are significantly different.
B. The means are equal.
C. We cannot conclude the means are different.
D. The sample sizes are insufficient.
Answer: C
Explanation: Since the interval includes zero, it suggests that the difference in means could be zero, so we cannot conclude a significant difference at 95% confidence.

Question 15:
What is the critical z-value for an 80% confidence interval?
A. 1.28
B. 1.96
C. 2.58
D. 1.645
Answer: A
Explanation: For an 80% confidence level, the critical z-value is 1.28, as it corresponds to the middle 80% of the standard normal distribution.

Question 16:
A confidence interval for a mean is calculated as (25, 35). If the confidence level increases to 99%, what happens to the interval?
A. It becomes narrower.
B. It stays the same.
C. It becomes wider.
D. It cannot be determined.
Answer: C
Explanation: Increasing the confidence level from 95% to 99% requires a larger critical value, making the interval wider based on the same data.

Question 17:
For a binomial proportion with p = 0.6 and n = 100, what is the 95% confidence interval?
A. (0.505, 0.695)
B. (0.500, 0.700)
C. (0.550, 0.650)
D. (0.600, 0.700)
Answer: A
Explanation: CI = 0.6 ± 1.96 * √[(0.6 * 0.4) / 100] = 0.6 ± 1.96 * 0.049 ≈ 0.6 ± 0.096, resulting in (0.504, 0.696), or approximately (0.505, 0.695).

Question 18:
If the standard error decreases, how does the confidence interval change?
A. It becomes wider.
B. It becomes narrower.
C. It remains the same.
D. It depends on the mean.
Answer: B
Explanation: A smaller standard error (due to larger sample size or smaller variability) reduces the margin of error, narrowing the confidence interval.

Question 19:
A 95% confidence interval for a standard deviation is based on a sample of 25 with s = 5. What is the interval?
A. (3.71, 8.05)
B. (4.00, 7.00)
C. (3.50, 6.50)
D. (2.50, 9.50)
Answer: A
Explanation: Using the chi-square distribution, for df=24, CI = √[( (n-1)s^2 / χ²_upper, √[( (n-1)s^2 / χ²_lower ] = √[(24*25) / 39.364] to √[(24*25) / 13.848] ≈ √[15.24] to √[33.05], resulting in approximately (3.71, 8.05).

Question 20:
What does a 95% confidence level mean for an interval?
A. The interval contains 95% of the data.
B. There is a 95% chance the true mean is in the interval.
C. 95% of intervals from repeated samples would contain the true mean.
D. The mean is 95% accurate.
Answer: C
Explanation: A 95% confidence level means that if we repeated the sampling process many times, about 95% of the calculated intervals would contain the true population parameter.