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

Question 1

Skill question

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.

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Question 2

Skill question

The following table shows the frequency of responses on a 5-point Likert scale, where 1 represents "Strongly Disagree" and 5 represents "Strongly Agree".

Score ( $x$ )	1	2	3	4	5
Frequency ( $f$ )	10	20	40	50	30

Convert the scores to a percentage scale where 1 maps to $0%$ and 5 maps to $100%$ . Calculate the mean percentage for the responses.

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Question 3

Skill question

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.

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Question 4

Skill question

Sample size calculation for comparing two proportions using the standard formula for power and significance.

Compute the required sample size per group to detect a difference between two proportions $p_{1} = 0.5$ and $p_{2} = 0.6$ with 80% power at a 5% significance level (two-sided).

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Question 5

Skill question

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

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Question 6

Skill question

This question assesses the ability to determine the minimum sample size required for estimating a population proportion with a specific margin of error and confidence level.

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 $p = 0.5$ .

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Question 7

Skill question

A finite population of size $N = 1000$ has a true proportion $p = 0.4$ . A simple random sample of size $n = 200$ is drawn without replacement. Calculate the variance of the sample proportion $\overset{p}{^}$ using the finite population correction.

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Question 8

Skill question

A survey of a population with a sample size of $n = 250$ resulted in a sample proportion of $\overset{p}{^} = 0.70$ .

Find a 95% confidence interval for the population proportion $p$ .

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Question 9

Skill question

Statistics - Discrete random variables - Discrete uniform distribution properties.

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

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Question 10

Skill question

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.

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Question 11

Skill question

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”?

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IB Mathematics Applications & Interpretation Question Type 2: Fixing the Line of Questioning for Poor Questions Exercises