Question: Consider the following boxplot of a quantitative variable measured on $n = 56$ 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 $[3, 4.5]$ is a 40% PLTI.
b. The interval $[1.5, 6.5]$ is a 90% PLTI.
c. The interval $[3, 4.5]$ is a 99% confidence interval for the population median.
d. The interval $[3, 4.5]$ is a 50% PI.
1. **Problem Statement:** We have a boxplot for a sample of size $n=56$ with whiskers approximately at $1.5$ and $6.5$, a box from $3$ to $4.5$, and a median near $3.5$. We need to determine which of the given statements about prediction intervals (PLTI), confidence intervals, and prediction intervals (PI) are true.
2. **Understanding the boxplot components:**
- The box represents the interquartile range (IQR), from the first quartile ($Q_1$) to the third quartile ($Q_3$), here approximately $3$ to $4.5$.
- The whiskers extend to the minimum and maximum values within $1.5 \times IQR$ from the quartiles, here approximately $1.5$ to $6.5$.
- The median is near $3.5$.
3. **Definitions:**
- A $p\%$ PLTI (Prediction Limit Tolerance Interval) is an interval that contains a specified proportion $p$ of the population with a certain confidence.
- A confidence interval for the population median estimates the median with a given confidence level.
- A prediction interval (PI) predicts where a future observation will fall with a certain probability.
4. **Evaluating each statement:**
**a. The interval $[3, 4.5]$ is a 40% PLTI.**
- The interval $[3, 4.5]$ corresponds to the IQR, which contains the middle 50% of the data.
- Since the IQR contains 50% of the data, it cannot be a 40% PLTI.
- **This statement is false.**
**b. The interval $[1.5, 6.5]$ is a 90% PLTI.**
- The whiskers extend approximately from $1.5$ to $6.5$, which typically cover about 99% of the data if there are no outliers.
- However, the whiskers here likely cover more than 90% of the data.
- So, the interval $[1.5, 6.5]$ contains more than 90% of the data.
- **This statement is true.**
**c. The interval $[3, 4.5]$ is a 99% confidence interval for the population median.**
- The IQR is not a confidence interval for the median; it is a measure of spread.
- Confidence intervals for the median are usually calculated using order statistics and are not simply the IQR.
- **This statement is false.**
**d. The interval $[3, 4.5]$ is a 50% PI.**
- The IQR contains 50% of the data, so it can be interpreted as a 50% prediction interval for a future observation.
- **This statement is true.**
5. **Final answers:**
- True statements: b and d.
- False statements: a and c.