MATH 225N WEEK 6 DISCUSSION: CONFIDENCE INTERVAL
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Foundations of Confidence Intervals (8 questions)
Q1: A nursing student tells you, "I calculated a 95% confidence interval for mean systolic
BP in my sample, and it came out to 118–126 mmHg. So there's a 95% probability that
the true population mean is somewhere between 118 and 126." What's wrong with this
statement?
A. Nothing—this is exactly what a 95% confidence interval means
B. The probability is actually 100%, not 95%, because the true mean must fall within the
calculated interval
C. The 95% refers to the reliability of the method over many samples, not the probability
that this particular interval captures the true mean; the true mean is fixed, so probability
doesn't apply to a single realized interval [CORRECT]
D. The student should have used a 99% confidence interval for a more accurate
probability statement
Rationale: The best answer is C. This is the classic confidence interval misconception.
Once you calculate an interval from your sample data, the true population mean either is
or isn't in that interval—there's no probability about it. The 95% refers to the long-run
,performance of the method: if you took 100 different samples and built 100 intervals,
about 95 of them would capture the true mean. It's about the process, not the product.
Correct Answer: C
Q2: Which of the following factors will make a confidence interval wider, assuming all
else stays the same?
A. Increasing the sample size from 50 to 200
B. Decreasing the standard deviation from 15 to 8
C. Increasing the confidence level from 90% to 99% [CORRECT]
D. Using a z-distribution instead of a t-distribution
Rationale: The best answer is C. Higher confidence means you want to be more "sure"
your interval captures the true parameter, so you need to cast a wider net. Think of it
this way: a 99% CI uses a larger critical value (z = 2.576) than a 90% CI (z = 1.645),
which directly increases the margin of error. Larger sample sizes and smaller standard
deviations both narrow the interval, and the t-distribution actually gives slightly wider
intervals than z for small samples because of the heavier tails.
Correct Answer: C
Q3: In a confidence interval formula, the margin of error represents:
A. The exact distance between the sample mean and the population mean
B. The maximum likely distance between the sample statistic and the population
parameter, based on the chosen confidence level and variability [CORRECT]
, C. The standard deviation of the sampling distribution
D. The probability that the interval contains the true parameter
Rationale: The best answer is B. Margin of error is the "fudge factor" that accounts for
the fact that your sample statistic probably isn't exactly equal to the population
parameter. It combines the critical value (how many standard errors you need for your
confidence level) with the standard error (how much sample means typically vary). It's
not the actual distance—that's unknown—it's the maximum likely distance given your
chosen confidence level.
Correct Answer: B
Q4: You are calculating a confidence interval for a population mean. The population
standard deviation is unknown, and your sample size is 18. Which distribution should
you use for the critical value?
A. Normal (z) distribution because n > 10
B. t-distribution with 17 degrees of freedom [CORRECT]
C. t-distribution with 18 degrees of freedom
D. Chi-square distribution because the standard deviation is unknown
Rationale: The best answer is B. When sigma is unknown and you're working with a
small sample (typically n < 30), you use the t-distribution to account for the extra
uncertainty in estimating sigma from your sample. Degrees of freedom for a
one-sample mean CI is always n – 1, so 18 – 1 = 17. The t-distribution has heavier tails
than the normal distribution, which makes the interval slightly wider—this is the
statistical penalty for not knowing the population standard deviation.
Actual Exam |UPDATED – Complete Q&A with Rationales –
Pass Guaranteed - A+ Graded
Foundations of Confidence Intervals (8 questions)
Q1: A nursing student tells you, "I calculated a 95% confidence interval for mean systolic
BP in my sample, and it came out to 118–126 mmHg. So there's a 95% probability that
the true population mean is somewhere between 118 and 126." What's wrong with this
statement?
A. Nothing—this is exactly what a 95% confidence interval means
B. The probability is actually 100%, not 95%, because the true mean must fall within the
calculated interval
C. The 95% refers to the reliability of the method over many samples, not the probability
that this particular interval captures the true mean; the true mean is fixed, so probability
doesn't apply to a single realized interval [CORRECT]
D. The student should have used a 99% confidence interval for a more accurate
probability statement
Rationale: The best answer is C. This is the classic confidence interval misconception.
Once you calculate an interval from your sample data, the true population mean either is
or isn't in that interval—there's no probability about it. The 95% refers to the long-run
,performance of the method: if you took 100 different samples and built 100 intervals,
about 95 of them would capture the true mean. It's about the process, not the product.
Correct Answer: C
Q2: Which of the following factors will make a confidence interval wider, assuming all
else stays the same?
A. Increasing the sample size from 50 to 200
B. Decreasing the standard deviation from 15 to 8
C. Increasing the confidence level from 90% to 99% [CORRECT]
D. Using a z-distribution instead of a t-distribution
Rationale: The best answer is C. Higher confidence means you want to be more "sure"
your interval captures the true parameter, so you need to cast a wider net. Think of it
this way: a 99% CI uses a larger critical value (z = 2.576) than a 90% CI (z = 1.645),
which directly increases the margin of error. Larger sample sizes and smaller standard
deviations both narrow the interval, and the t-distribution actually gives slightly wider
intervals than z for small samples because of the heavier tails.
Correct Answer: C
Q3: In a confidence interval formula, the margin of error represents:
A. The exact distance between the sample mean and the population mean
B. The maximum likely distance between the sample statistic and the population
parameter, based on the chosen confidence level and variability [CORRECT]
, C. The standard deviation of the sampling distribution
D. The probability that the interval contains the true parameter
Rationale: The best answer is B. Margin of error is the "fudge factor" that accounts for
the fact that your sample statistic probably isn't exactly equal to the population
parameter. It combines the critical value (how many standard errors you need for your
confidence level) with the standard error (how much sample means typically vary). It's
not the actual distance—that's unknown—it's the maximum likely distance given your
chosen confidence level.
Correct Answer: B
Q4: You are calculating a confidence interval for a population mean. The population
standard deviation is unknown, and your sample size is 18. Which distribution should
you use for the critical value?
A. Normal (z) distribution because n > 10
B. t-distribution with 17 degrees of freedom [CORRECT]
C. t-distribution with 18 degrees of freedom
D. Chi-square distribution because the standard deviation is unknown
Rationale: The best answer is B. When sigma is unknown and you're working with a
small sample (typically n < 30), you use the t-distribution to account for the extra
uncertainty in estimating sigma from your sample. Degrees of freedom for a
one-sample mean CI is always n – 1, so 18 – 1 = 17. The t-distribution has heavier tails
than the normal distribution, which makes the interval slightly wider—this is the
statistical penalty for not knowing the population standard deviation.
