Estimation & Confidence Concepts
A study reports an average salary of $48,000 with no range attached.
What type of estimate is this?
POINT ESTIMATE.
A point estimate is a single sample statistic used to estimate a population parameter.
s = 25, N = 100
What is √N?
√100 = 10
Out of 250 students, 50 report owning a car.
Calculate the sample proportion (p).
p = f / N = 50 / 250 = 0.20
20% of students in the sample own a car.
The population mean is 60.
Write the null hypothesis.
H₀: μ = 60
The null hypothesis always includes equality; it states the population mean equals the specified value.
In a bivariate table, which variable goes in the columns?
The INDEPENDENT VARIABLE (IV) goes in the columns.
The dependent variable (DV) goes in the rows.
Rule: IV - columns, DV - rows.
Which confidence interval is more precise:
(40, 60) or (48, 52)?
Explain how you know.
(48, 52) is more precise — it is the narrower interval.
Width of (40, 60) = 20. Width of (48, 52) = 4.
s = 18, N = 81
Calculate the standard error.
sᵧ = s / √N = 18 / √81 = 18 / 9 = 2
The standard error is 2.
If p = 0.35, what is (1 − p)?
Why do we need this value for the standard error formula?
1 − p = 1 − 0.35 = 0.65
We need (1 − p) because the standard error formula for proportions is:
sₚ = √[ p(1 − p) / N ]
Both p and (1 − p) together reflect how split the sample is.
A researcher predicts that the true mean is LOWER than 60.
Write the research hypothesis.
H₁: μ < 60
This is a one-tailed (left-tailed) test because the prediction is directional, specifically that the mean is lower than 60.
Why do researchers use column percentages instead of raw counts when comparing groups in a bivariate table?
To compare groups FAIRLY — especially when group sizes differ.
Raw counts can be misleading if one group is much larger than another.
Column percentages standardize the comparison so we can say 'X% of Group A vs. X% of Group B.'
If sample size increases from 25 to 100, what happens to the confidence interval?
Explain why using the standard error formula.
The confidence interval becomes NARROWER — more precise.
SE = 3, Z = 1.96
Calculate the margin of error.
Margin of error = Z × SE = 1.96 × 3 = 5.88
p = 0.50, N = 100
Calculate the standard error of the proportion.
sₚ = √[ p(1 − p) / N ]
= √[ 0.50(0.50) / 100 ]
= √[ 0.25 / 100 ]
= √0.0025
= 0.05
σ = 12, N = 36
Calculate the standard error.
SE = σ / √N = 12 / √36 = 12 / 6 = 2
The standard error is 2.
Group A: 30 out of 60 responded Yes.
Group B: 20 out of 40 responded Yes.
Calculate the column percentage for each group.
Is there a relationship?
Group A: 30 / 60 × 100 = 50%
Group B: 20 / 40 × 100 = 50%
NO relationship — the column percentages are identical (50% = 50%).
A relationship exists only when percentages DIFFER across IV categories.
If the confidence level increases from 90% to 99%, what happens to the interval?
Explain why using the Z values.
The confidence interval becomes WIDER — less precise.
90% - Z = 1.65 99% - Z = 2.58
A higher confidence level requires a larger Z value.
Mean = 65, SE = 2
Construct a 95% confidence interval.
ME = Z × SE = 1.96 × 2 = 3.92
CI = 65 ± 3.92
95% CI = (61.08 , 68.92)
p = 0.40, SE = 0.05
Calculate the margin of error at a 95% confidence level.
ME = Z × sₚ = 1.96 × 0.05 = 0.098
The margin of error is approximately ±0.10 (or ±10%).
Ȳ = 66, μ = 60, SE = 3
Calculate the Z statistic.
Z = (Ȳ − μ) / SE = (66 − 60) / 3 = 6 / 3 = 2
The Z obtained is 2.
Group A: 45% support
Group B: 70% support
Is there a relationship? How strong is it?
YES — there is a relationship.
The column percentages differ by 25 percentage points (70% − 45% = 25%).
Rule: A relationship exists when column percentages differ across groups.
A 25-point difference indicates a moderate-to-strong relationship.
Two studies estimate the same population mean and use the same sample size.
Study A has SD = 10. Study B has SD = 20.
Which study will produce a wider confidence interval, and why?
Study B will produce the WIDER interval.
Mean = 72, s = 24, N = 144
Construct a 95% confidence interval and interpret it in one sentence.
SE = s / √N = 24 / √144 = 24 / 12 = 2
ME = 1.96 × 2 = 3.92
CI = 72 ± 3.92
95% CI = (68.08 , 75.92)
Interpretation: We are 95% confident the true population mean falls between 68.08 and 75.92.
p = 0.30, N = 100
Construct a 95% confidence interval and interpret it in one sentence.
sₚ = √[ 0.30(0.70) / 100 ] = √[ 0.21 / 100 ] = √0.0021 ≈ 0.046
ME = 1.96 × 0.046 ≈ 0.090
CI = 0.30 ± 0.090
95% CI = (0.210 , 0.390)
Interpretation: We are 95% confident the true population proportion falls between 21.0% and 39.0%.
Ȳ = 52, μ = 50, σ = 10, N = 25
α = 0.05 (two-tailed) Z critical = ±1.96
Show all work, make a decision, and write a one-sentence conclusion.
SE = σ / √N = 10 / √25 = 10 / 5 = 2
Z = (Ȳ − μ) / SE = (52 − 50) / 2 = 2 / 2 = 1
Decision: FAIL TO REJECT H₀
Z obtained (1.00) < Z critical (1.96) — does not fall in the rejection region.
Conclusion: There is not sufficient evidence to conclude that the population mean differs from 50.
A bivariate table shows:
Group A = 65% support a new policy
Group B = 30% support the same policy
Interpret the relationship fully — existence, strength, and direction.
Existence: YES — a relationship exists. The percentages differ by 35 points.
Strength: STRONG — a 35-point gap indicates a strong relationship.
The larger the percentage difference, the stronger the relationship.
Direction: Group A is much more likely to support the policy (65%) than Group B (30%). As you move from Group B to Group A, support increases.