Question: Consider the following boxplot of a quantitative variable measured on $n = 48$ cases which we selected by simple random sampling from some population of interest:
Select each of the true statements from the following:
a. The interval $[3.5, 4.25]$ is a 40% PI.
b. The interval $[3, 4.25]$ is a 90% PLTI.
c. The interval $[3, 4.25]$ is a 99% confidence interval for the population median.
d. The interval $[3, 3.5]$ is a 50% PI.
Graph: a boxplot near the center of the page with whiskers extending to about $1.5$ on the left and $5.5$ on the right, a box from about $3$ to $4.25$ with median near $3.5$, and outliers at about $-3$ and $6.5$; position_hint = center
1. **State the problem:** We are given a boxplot of a quantitative variable with $n=48$ cases and asked to determine which of the given interval statements are true.
2. **Understand the terms:**
- PI = Prediction Interval: interval likely to contain a future observation.
- PLTI = Probably Likely True Interval (less common, but here likely refers to an interval containing a certain proportion of the data).
- Confidence Interval for the population median: interval estimate for the median of the population.
3. **Analyze the boxplot:**
- Whiskers extend approximately from $1.5$ to $5.5$.
- Box extends from $3$ (Q1) to $4.25$ (Q3).
- Median is near $3.5$.
- Outliers at about $-3$ and $6.5$.
4. **Evaluate each statement:**
**a. The interval $[3.5, 4.25]$ is a 40% PI.**
- The boxplot's interquartile range (IQR) is $[3, 4.25]$ which contains 50% of the data.
- The interval $[3.5, 4.25]$ is only part of the upper half of the box, less than 50%, so it cannot be a 40% PI.
- Prediction intervals usually cover more than 50% of data or future observations.
- **Conclusion:** False.
**b. The interval $[3, 4.25]$ is a 90% PLTI.**
- The box $[3, 4.25]$ is the IQR containing 50% of the data.
- 90% PLTI would be wider, likely including whiskers.
- So $[3, 4.25]$ is not 90% but 50% interval.
- **Conclusion:** False.
**c. The interval $[3, 4.25]$ is a 99% confidence interval for the population median.**
- Confidence intervals for the median are usually estimated using order statistics and are wider than the IQR.
- The IQR is not a confidence interval for the median.
- **Conclusion:** False.
**d. The interval $[3, 3.5]$ is a 50% PI.**
- The IQR is $[3, 4.25]$ which contains 50% of the data.
- $[3, 3.5]$ is only part of the lower half of the box, less than 50%.
- So it cannot be a 50% PI.
- **Conclusion:** False.
**Summary:** None of the statements a, b, c, or d are true based on the boxplot and definitions.
5. **Final answer:** No statements are true.