A random sample of n = 25 has mean x-bar = 72, and the population standard deviation is known to be sigma = 10. Using z = 1.96, what is the 95% confidence interval for mu, rounded to two decimals?
(68.08, 75.92)
✓
(52.40, 91.60)
(71.22, 72.78)
(68.71, 75.29)
Correct Answer
(68.08, 75.92)
The error bound is 1.96 x 10 / sqrt(25) = 1.96 x 2 = 3.92, so the interval is 72 - 3.92 to 72 + 3.92, or (68.08, 75.92). The interval (52.40, 91.60) uses sigma instead of sigma / sqrt(n). The interval (71.22, 72.78) divides sigma by n instead of by the square root of n. The interval (68.71, 75.29) uses 1.645, the critical value for 90% confidence.
Category: Error Bound
Question 2
A confidence interval for a population mean runs from 40 to 52. What is the error bound for the mean, EBM?
12
6
✓
46
3
Correct Answer
6
The interval is symmetric about the sample mean, so the error bound is half the width: (52 - 40) / 2 = 6. The value 12 is the full width of the interval. The value 46 is the sample mean, the center of the interval. The value 3 is only a quarter of the width.
Category: Error Bound
Question 3
In a random sample of 200 adults, 80 say they exercise daily. Using z = 1.96, what is the 95% confidence interval for the population proportion, rounded to three decimals?
(0.343, 0.457)
(0.395, 0.405)
(0.332, 0.468)
✓
(0.532, 0.668)
Correct Answer
(0.332, 0.468)
Here p' = 80 / 200 = 0.40, and the error bound is 1.96 x sqrt(0.40 x 0.60 / 200) = 1.96 x 0.0346 = 0.068, giving (0.332, 0.468). The interval (0.343, 0.457) uses 1.645, the 90% critical value. The interval (0.395, 0.405) divides sqrt(p'q') by n instead of by the square root of n. The interval (0.532, 0.668) is centered at q' = 0.60, the proportion who do not exercise daily.
Category: Population Proportion
Question 4
A researcher computes a 95% confidence interval for mu. Which statement correctly describes what the 95% means?
95% of the sample values lie inside this interval
The sample mean has a 95% chance of lying in this interval
Mu moves into this interval 95% of the time
About 95% of intervals built this way would capture mu
✓
Correct Answer
About 95% of intervals built this way would capture mu
The confidence level is the long-run success rate of the method: if many samples were taken and an interval built from each, about 95% of those intervals would contain the true mean. The claim that 95% of the sample values lie inside this interval confuses an interval for mu with a range for individual data. The claim that the sample mean has a 95% chance of lying in this interval is wrong because x-bar is always at the center. The claim that mu moves into this interval 95% of the time is wrong because mu is a fixed number that does not move.
Category: Interpreting Confidence Intervals
Question 5
Which z critical value is used for a 99% confidence interval for a mean when sigma is known?
2.576
✓
1.96
1.645
2.326
Correct Answer
2.576
For 99% confidence, 0.005 of the area lies in each tail, and the z value that leaves 0.005 above it is 2.576. The value 1.96 is used for 95% confidence. The value 1.645 is used for 90% confidence. The value 2.326 is used for 98% confidence.
Category: Width and Sample Size
Question 6
A researcher wants to estimate a mean to within 3 units with 95% confidence, and sigma = 15. Using z = 1.96 and rounding up to the next whole number, what sample size is needed?
96
97
✓
10
68
Correct Answer
97
The sample size formula gives n = (1.96 x 15 / 3)^2 = 9.8^2 = 96.04, and sample sizes are always rounded up, so n = 97. The value 96 rounds down, which would leave the error bound slightly above 3. The value 10 forgets to square 9.8. The value 68 uses 1.645, the 90% critical value.
Category: Width and Sample Size
Question 7
A confidence interval is built with a 90% confidence level. What is alpha?
0.05
0.90
0.10
✓
0.01
Correct Answer
0.10
Alpha is the probability that the method produces an interval that misses the parameter, and alpha plus the confidence level equals 1, so alpha = 1 - 0.90 = 0.10. The value 0.05 is alpha / 2, the area in each tail. The value 0.90 is the confidence level itself. The value 0.01 is alpha for a 99% confidence level.
Category: Interpreting Confidence Intervals
Question 8
In the error bound EBM = z x sigma / sqrt(n), the value of z is determined by which choice?
The sample mean
The sample size
The population mean
The confidence level
✓
Correct Answer
The confidence level
The critical value z is the standard normal value that leaves alpha / 2 in each tail, so it depends only on the confidence level chosen. The sample mean is the center of the interval and does not affect z. The sample size affects the error bound through sqrt(n), not through z. The population mean is the unknown being estimated.
Category: Error Bound
Question 9
A random sample of 15 measurements comes from a normally distributed population whose standard deviation is unknown. Which distribution supplies the critical value for a confidence interval for mu?
Student's t with 14 degrees of freedom
✓
Student's t with 15 degrees of freedom
The standard normal distribution
Student's t with 16 degrees of freedom
Correct Answer
Student's t with 14 degrees of freedom
When sigma is unknown and the sample standard deviation s is used, the critical value comes from the t-distribution with n - 1 = 14 degrees of freedom. Student's t with 15 degrees of freedom uses n instead of n - 1. The standard normal distribution is used when sigma is known. Student's t with 16 degrees of freedom uses n + 1, which has no basis.
Category: Student's t-Distribution
Question 10
Compared with the z-distribution, how does a t-distribution with few degrees of freedom differ?
It has less probability in its tails
It has more probability in its tails
✓
It has a mean greater than zero
It has a strong right skew
Correct Answer
It has more probability in its tails
A t-distribution is more spread out than the standard normal, with more probability in the tails and less in the center, which reflects the extra uncertainty from estimating sigma with s. Less probability in its tails reverses the difference. A mean greater than zero is wrong because a t-distribution is centered at zero. A strong right skew is wrong because the t-distribution is symmetric.
Category: Student's t-Distribution
Question 11
A sample of n = 64 comes from a population with known sigma = 16, and x-bar = 100. Using z = 1.645, what is the error bound for a 90% confidence interval for mu?
26.32
3.92
3.29
✓
0.41
Correct Answer
3.29
The standard error is 16 / sqrt(64) = 2, so the error bound is 1.645 x 2 = 3.29. The value 26.32 multiplies 1.645 by sigma instead of by the standard error. The value 3.92 uses 1.96, the 95% critical value. The value 0.41 divides sigma by n instead of by the square root of n.
Category: Error Bound
Question 12
A sample of 16 has standard deviation s = 8, and sigma is unknown. With t = 2.131 for 95% confidence and 15 degrees of freedom, what is the error bound for the mean, rounded to two decimals?
17.05
1.07
3.92
4.26
✓
Correct Answer
4.26
With sigma unknown, the error bound is t x s / sqrt(n) = 2.131 x 8 / 4 = 2.131 x 2 = 4.26. The value 17.05 multiplies t by s without dividing by sqrt(n). The value 1.07 divides s by n instead of by the square root of n. The value 3.92 uses z = 1.96, which is not appropriate when s replaces sigma.
Category: Student's t-Distribution
Question 13
In a survey of 400 voters, 120 support a ballot measure. What is the point estimate p' of the population proportion who support it?
0.30
✓
0.70
0.023
0.43
Correct Answer
0.30
The point estimate is p' = x / n = 120 / 400 = 0.30. The value 0.70 is q', the proportion who do not support it. The value 0.023 is the estimated standard deviation of p', sqrt(0.30 x 0.70 / 400). The value 0.43 divides supporters by nonsupporters, 120 / 280, instead of by the sample size.
Category: Population Proportion
Question 14
A sample of n = 100 gives p' = 0.5. Using z = 1.96, what is the error bound for the proportion, EBP?
0.082
0.098
✓
0.05
0.196
Correct Answer
0.098
The error bound is z x sqrt(p'q' / n) = 1.96 x sqrt(0.25 / 100) = 1.96 x 0.05 = 0.098. The value 0.082 uses 1.645, the 90% critical value. The value 0.05 is the standard deviation of p' before multiplying by z. The value 0.196 is twice the error bound, which is the width of the interval.
Category: Population Proportion
Question 15
A poll finds that 52% of a random sample favor a candidate, with a 95% error bound of 3 percentage points. What is the confidence interval for the population proportion?
(0.50, 0.54)
(0.52, 0.55)
(0.49, 0.55)
✓
(0.46, 0.58)
Correct Answer
(0.49, 0.55)
The interval is the point estimate plus or minus the error bound: 0.52 - 0.03 to 0.52 + 0.03, or (0.49, 0.55). The interval (0.50, 0.54) uses an error bound of 2 points instead of 3. The interval (0.52, 0.55) adds the error bound but never subtracts it. The interval (0.46, 0.58) uses 6 points, double the error bound.
Category: Population Proportion
Question 16
With no prior estimate of p, a planner uses p' = 0.5 to estimate a proportion within 0.05 at 95% confidence. Using z = 1.96 and rounding up, how many people must be sampled?
384
20
271
385
✓
Correct Answer
385
The formula gives n = 1.96^2 x 0.5 x 0.5 / 0.05^2 = 3.8416 x 0.25 / 0.0025 = 384.16, which rounds up to 385. The value 384 rounds down, which would not quite meet the target error bound. The value 20 forgets to square, using 1.96 x 0.5 / 0.05 = 19.6. The value 271 uses 1.645, the 90% critical value.
Category: Width and Sample Size
Question 17
In a sample of 25 students, 6 own a tablet. Using the plus-four method, what proportion is used as the point estimate, rounded to three decimals?
0.276
✓
0.240
0.320
0.207
Correct Answer
0.276
The plus-four method adds two successes and two failures, so the estimate is (6 + 2) / (25 + 4) = 8 / 29 = 0.276. The value 0.240 is the ordinary estimate 6 / 25 with no adjustment. The value 0.320 adds 2 to the successes but leaves n at 25. The value 0.207 adds 4 to n but leaves the successes at 6.
Category: Plus-Four Method
Question 18
Two researchers use the same sample size and confidence level, but one population has a larger standard deviation. How does that researcher's confidence interval for mu compare?
It is narrower
It is wider
✓
It has the same width
It is shifted toward zero
Correct Answer
It is wider
The error bound z x sigma / sqrt(n) grows in proportion to sigma, so a more variable population gives a wider interval. It is narrower reverses the effect. It has the same width ignores that sigma appears in the error bound. It is shifted toward zero confuses the center of the interval, which is x-bar, with its width.
Category: Width and Sample Size
Question 19
When a t-interval for mu is built from a small sample, what is assumed about the population the data come from?
It is strongly skewed
It is uniformly distributed
It is approximately normal
✓
It is exponentially distributed
Correct Answer
It is approximately normal
The t procedures assume a simple random sample from a population that is normal or approximately normal, which matters most when the sample is small. A strongly skewed population violates that assumption. A uniformly distributed population is not the assumed model. An exponentially distributed population is skewed and also not the assumed model.
Category: Student's t-Distribution
Question 20
In a confidence interval for a population mean, which quantity serves as the point estimate?
The error bound EBM
The confidence level
The population standard deviation
The sample mean x-bar
✓
Correct Answer
The sample mean x-bar
The point estimate of mu is the sample mean x-bar, the single value at the center of the interval. The error bound EBM is the distance added to and subtracted from the point estimate. The confidence level describes how reliable the method is. The population standard deviation measures spread and is used to compute the error bound.
Category: Point Estimates