Question Type 2: Fixing the line of questioning for poor questions Exercises

In a survey of 250 respondents to a closed yes/no question, 175 answered “yes”. Calculate the sample proportion $\hat p$ of “yes” responses and the variance of $\hat p$.

A poorly worded survey question yielded 70% “agree” from 400 respondents. After rewriting to remove bias, only 52% agreed out of another 400 respondents. Calculate the absolute and relative change in agreement rate.

Convert the 5-point Likert data in the previous question to a percentage scale where 1 maps to 0% and 5 maps to 100%. Then compute the new mean percentage.

In a random sample of 5 survey respondents from a large population where the true “yes” probability is 0.6, what is the probability exactly 3 answer “yes”?

A 5-point Likert question yielded the following counts: 10 respondents chose 1, 20 chose 2, 40 chose 3, 50 chose 4 and 30 chose 5. Compute the sample mean and sample standard deviation.

Using the result $\hat p = 0.70$ from the previous question, find a 95% confidence interval for the true proportion $p$.

A survey uses a question on a 4-point scale from “Strongly Disagree” to “Strongly Agree”. If the true underlying opinions follow a uniform distribution over $\{1,2,3,4\}$, compute the expected mean and variance of the ratings.

Determine the minimum sample size required to estimate a population proportion within a margin of error of 3% at 95% confidence, assuming a conservative estimate $p=0.5$.

A finite population of size $N=1000$ has a true proportion $p=0.4$. You draw a simple random sample of size $n=200$ without replacement. Compute the variance of the sample proportion $\hat p$ using the finite population correction.

Two survey question wordings yield these results: Sample A (biased wording): 80 out of 150 “yes”; Sample B (neutral wording): 60 out of 150 “yes”. Test at the 5% level whether the proportions differ significantly.

Compute the required sample size to detect a difference between two proportions $p_1=0.5$ and $p_2=0.6$ with 80% power at 5% significance (two-sided).