LQ & Prob, Z
An LQ of 1.0 indicates what?
The local share equals the reference-region share.
What is the main difference between a z-distribution and a t-distribution?
The t-distribution has heavier tails, with shape depending on degrees of freedom; it approaches the normal distribution as degrees of freedom increase.
Which sampling method gives every population member an equal probability of selection?
Simple random sampling.
What does the null hypothesis generally represent?
A statement of no difference, no effect, or no association
What does a one-way ANOVA compare?
Means across three or more independent groups (though it can also be used with two).
A county has 12% employment in manufacturing, compared with 8% nationally. Calculate its LQ.
LQ = 12/8 = 1.50. Manufacturing is 50% more concentrated locally.
What happens to the standard error when sample size increases from 25 to 100, holding standard deviation constant?
It is cut in half, because SE = s/√n.
A researcher divides a city into low-, medium-, and high-income neighborhoods and randomly samples households within each group. Identify the method.
Stratified random sampling.
A researcher asks whether mean tree canopy differs from 20%. Write H₀ and Hₐ. Is it one or two tailed?
H₀: μ = 20%; Hₐ: μ ≠ 20%. This is two-tailed.
A researcher examines whether land-use type is associated with flood-risk category. Which test is appropriate?
Chi-squared test of independence.
Part 1: If P(A)=0.30 and P(B)=0.20 and events are mutually exclusive, find P(A or B).
Part 2: If P(A)=0.4 and P(B)=0.5 and the events are independent, find P(A and B).
Part 1: 0.50, using the addition rule for mutually exclusive events.
Part 2: 020
A sample mean is 60 and the margin of error is 5. What is the confidence interval?
55 to 65
A researcher randomly selects five census tracts and surveys every household within those tracts. Identify the sampling method.
One-stage cluster sampling.
If p = 0.03 and α = 0.05, what is the statistical decision?
Reject H₀ because p < α.
A one-way ANOVA returns p = 0.02 at α = 0.05. What can the researcher conclude?
Reject H₀. At least one population mean differs, but ANOVA alone does not identify which groups differ.
A neighborhood's mean summer temperature is 32°C, compared with a regional mean of 28°C and standard deviation of 2°C. Calculate the z-score.
z = (32−28)/2 = +2. The neighborhood is two standard deviations above the regional mean.
A sample has n = 16, mean = 40, and s = 8. Calculate the standard error and degrees of freedom for a one-sample t interval.
SE = 8/√16 = 2; df = 15.
Explain what a 95% confidence level means.
Over many repetitions of the sampling procedure, approximately 95% of intervals constructed this way would contain the true population parameter.
A researcher compares the same 20 stream sites before and after restoration. Which t-test is appropriate, and why?
Paired-samples t-test, because observations are matched by site.
In a contingency table, the row total is 40, the column total is 30, and the grand total is 120. Calculate the expected frequency.
Expected frequency = (40×30)/120 = 10.
For a standard normal distribution, what is the probability that Z is between −1.00 and +1.00?
Approximately 0.6827, or 68.27%.
For a 95% t confidence interval, sample mean = 50, SE = 2, and critical t = 2.10. Calculate the interval.
50 ± (2.10×2) = 45.8 to 54.2.
Two studies have the same sample mean and standard deviation, but one has n=25 and the other n=100. Which has the narrower 95% CI, assuming independent observations and the same method?
The n=100 study, because it has a smaller standard error and typically a smaller critical value.
For a two-tailed test, the calculated t = −2.75, and the critical values are ±2.10. What is the conclusion?
Reject H₀ because |−2.75| > 2.10.
For a 3×4 contingency table, calculate the chi-squared degrees of freedom. If χ² = 14.0 and the critical value is 12.59, what is the decision?
df = (3−1)(4−1) = 6. Reject H₀ because 14.0 > 12.59.