ELEMENTARY STATISTICS EXAM 4 STUDY
GUIDE FULL QUESTIONS WITH DETAILED
ANSWERS
●● What is margin of error and what does it tell us about a point
estimate?
Answer: Margin of error is the maximum expected difference between
the sample estimate and the true population value.It tells us how "off"
our estimate could reasonably be.
●● When can we use a z-confidence interval for estimating μ?
Answer: When:
The sample is random
Population standard deviation (σ) is known
The population is normal OR the sample size is large This allows us to
use the standard normal (z) distribution.
●● What is a confidence level (c) and how does it relate to z-values?
Answer: The confidence level is the probability that the interval will
contain the true mean μ. It corresponds to the area under the standard
normal curve between −zc and +zc. Common values:
90% → 1.645
95% → 1.96
,99% → 2.58
●● How do you find the critical value zc from α (significance level)?
Answer: First, α = 1 − c
Then divide by 2 → α/2
Find the z-value with area = 1 − α/2 in the z-table Example: 95% → α =
0.05 → α/2 = 0.025 → z = 1.96
●● What is the margin of error formula and what does it mean?
Answer: E = zc × (σ / √n)
zc = confidence level
σ = variability
n = sample size Larger n → smaller error; higher confidence → larger
error
●● What is the confidence interval for μ and how do you interpret it?
Answer: Confidence interval = x̄ ± zc × (σ / √n)Interpretation: If we
repeat sampling, about c% of intervals will contain μ.
●● What is the confidence interval formula for μ when σ is known?
Answer: CI = (x̄ − E, x̄ + E) where E = zc × (σ / √n) So full formula: x̄ ±
zc × (σ / √n)
, ●● what does each part of the CI formula represent?
Answer: x̄ = sample mean (center of interval)
zc = confidence level (controls width)
σ = population variability
n = sample size
E = margin of error (distance from center)
●● What are the steps to construct a confidence interval?
Answer: Identify x̄, n, σ (or s), and confidence level c
Find zc from table
Compute margin of error: E = zc × (σ / √n)
Compute interval: (x̄ − E, x̄ + E)
Interpret the result
●● How do you interpret a confidence interval?
Answer: "We are c% confident that the true population mean μ lies
between [lower bound, upper bound]."
●● EXAMPLE
Given x̄ = 31.39, σ = 0.8, n = 82, 95% CI — what are the key steps? A
nswer: zc = 1.96
E = 1.96 × (0.8 / √82) ≈ 0.17
GUIDE FULL QUESTIONS WITH DETAILED
ANSWERS
●● What is margin of error and what does it tell us about a point
estimate?
Answer: Margin of error is the maximum expected difference between
the sample estimate and the true population value.It tells us how "off"
our estimate could reasonably be.
●● When can we use a z-confidence interval for estimating μ?
Answer: When:
The sample is random
Population standard deviation (σ) is known
The population is normal OR the sample size is large This allows us to
use the standard normal (z) distribution.
●● What is a confidence level (c) and how does it relate to z-values?
Answer: The confidence level is the probability that the interval will
contain the true mean μ. It corresponds to the area under the standard
normal curve between −zc and +zc. Common values:
90% → 1.645
95% → 1.96
,99% → 2.58
●● How do you find the critical value zc from α (significance level)?
Answer: First, α = 1 − c
Then divide by 2 → α/2
Find the z-value with area = 1 − α/2 in the z-table Example: 95% → α =
0.05 → α/2 = 0.025 → z = 1.96
●● What is the margin of error formula and what does it mean?
Answer: E = zc × (σ / √n)
zc = confidence level
σ = variability
n = sample size Larger n → smaller error; higher confidence → larger
error
●● What is the confidence interval for μ and how do you interpret it?
Answer: Confidence interval = x̄ ± zc × (σ / √n)Interpretation: If we
repeat sampling, about c% of intervals will contain μ.
●● What is the confidence interval formula for μ when σ is known?
Answer: CI = (x̄ − E, x̄ + E) where E = zc × (σ / √n) So full formula: x̄ ±
zc × (σ / √n)
, ●● what does each part of the CI formula represent?
Answer: x̄ = sample mean (center of interval)
zc = confidence level (controls width)
σ = population variability
n = sample size
E = margin of error (distance from center)
●● What are the steps to construct a confidence interval?
Answer: Identify x̄, n, σ (or s), and confidence level c
Find zc from table
Compute margin of error: E = zc × (σ / √n)
Compute interval: (x̄ − E, x̄ + E)
Interpret the result
●● How do you interpret a confidence interval?
Answer: "We are c% confident that the true population mean μ lies
between [lower bound, upper bound]."
●● EXAMPLE
Given x̄ = 31.39, σ = 0.8, n = 82, 95% CI — what are the key steps? A
nswer: zc = 1.96
E = 1.96 × (0.8 / √82) ≈ 0.17
