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% PLTI.
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: boxplot centered in the middle-right of the page on a horizontal number line from -3 to 7, with an outlier near 0 on the left and an outlier near 7 on the right; box spans about 3 to 4.25 with median around 3.5, and whiskers extend to about 2 and 5.5; position_hint=top-right
makng statistical inferences from ferrum college textbook for mat 208 introductory statistics
1. **Problem Statement:**
We are given a boxplot for a quantitative variable with $n=48$ cases and asked to determine which of the given intervals correspond to specific types of statistical intervals: PLTI (Percentile Likelihood Tolerance Interval), confidence interval for the median, or prediction interval (PI).
2. **Understanding the Boxplot and Intervals:**
- The boxplot shows the minimum, first quartile (Q1), median, third quartile (Q3), and maximum (excluding outliers).
- The box spans approximately from $3$ (Q1) to $4.25$ (Q3).
- The median is about $3.5$.
- Whiskers extend roughly from $2$ to $5.5$.
- Outliers are near $0$ and $7$.
3. **Definitions and Important Rules:**
- A **PLTI** (Percentile Likelihood Tolerance Interval) covers a specified percentage of the population with a certain confidence.
- A **confidence interval for the median** estimates the population median with a given confidence level.
- A **prediction interval (PI)** predicts where a future observation will fall with a certain probability.
4. **Analyzing Each Statement:**
**a. The interval $[3.5, 4.25]$ is a 40% PLTI.**
- The interval $[3.5, 4.25]$ lies between the median and Q3.
- The interquartile range (IQR) is $[3, 4.25]$, which covers 50% of the data.
- The interval $[3.5, 4.25]$ covers less than half the data, roughly 25% (from median to Q3).
- So, it cannot be a 40% PLTI.
**b. The interval $[3, 4.25]$ is a 90% PLTI.**
- The interval $[3, 4.25]$ is the IQR, which covers 50% of the data.
- 90% PLTI would be wider than the IQR.
- So, this is false.
**c. The interval $[3, 4.25]$ is a 99% confidence interval for the population median.**
- Confidence intervals for the median are often constructed using order statistics.
- The IQR is not a confidence interval for the median but a measure of spread.
- So, this is false.
**d. The interval $[3, 3.5]$ is a 50% PI.**
- Prediction intervals predict where a future observation will fall.
- The interval $[3, 3.5]$ covers from Q1 to median, about 25% of data.
- 50% PI would be wider, likely the IQR.
- So, this is false.
5. **Summary:**
- None of the statements a, b, c, or d are true based on the boxplot and definitions.
**Final answer:** None of the given statements are true.