The Environmental Protection Agency(EPA) uses a measure called the Pollutant Standards Index(PSI) to measure air quality in a city. A PSI reading over 100 indicates a day when the air quality is considered unhealthy. The measurements represent the number of days in 1995 that the PSI was over 100 for metropolitan areas in the U.S. Midwest. Summary statistics are provided. Which of the following statements is true?

min Q1 median Q3 max mean sd n
0 1 2.5 4.25 14 3.1 3.747 20

Question

The Environmental Protection Agency(EPA) uses a measure called the Pollutant Standards Index(PSI) to measure air quality in a city. A PSI reading over 100 indicates a day when the air quality is considered unhealthy. The measurements represent the number of days in 1995 that the PSI was over 100 for metropolitan areas in the U.S. Midwest. Summary statistics are provided. Which of the following statements is true?

min	Q1	median	Q3	max	mean	sd	n 
0	1	2.5	4.25	14	3.1	3.747	20

BlackTom AI · Accepted Answer

The prompt provides a question about interpreting a summary statistics table for PSI over 100 days across metropolitan areas in the U.S. Midwest in 1995, followed by four answer choices. We will evaluate each option based on the given statistics: min = 0, Q1 = 1, median = 2.5, Q3 = 4.25, max = 14, mean = 3.1, standard deviation = 3.747, n = 20.

Option 1: 'there is evidence that this distribution is left-skewed.' A left-skewed (negatively skewed) distribution would have a longer left tail and typically a mean less than the median. Here, the mean (3.1) is slightly greater than the median (2.5), which suggests a slight right (positive) skew rather than left skew. Additionally, the max is 14 while the min is 0, which can be consistent with some right skew. Therefore, the data appear more consistent with right skew than left skew, making this statement less plausible.

Option 2: '25% of metropolitan areas reported 0 or 1 days in which the PSI was over 100.' To check this, note the quartiles: Q1 = 1, and the minimum is 0. By definition, 25% of observations lie at or below Q1. If Q1 equals 1, that means 25% of the data are at or below 1. Since the minimum is 0, and the first quartile is 1, it is consistent that 25% of metropolitan areas had 0 or 1 days. This aligns with the quartile information and the typical interpretation of Q1.

Option 3: '14 metropolitan areas participated in this study.' The sample size n is given as 20 in the table. Therefore, claiming that only 14 participated contradicts the provided n and is incorrect.

Option 4: 'the EPA should not be concerned because there were no PSI measurements exceeding 100.' The max is 14, which means there were PSI measurements exceeding 100 on some days (at least up to 14 days for some metro areas), so this statement is false. It would be inappropriate to claim no measurements exceeded 100 when the max is 14 and the spread includes 0 and 14.

In summary, option 2 correctly interprets the quartile information to indicate that 25% of metropolitan areas had 0 or 1 day with PSI over 100, while the other statements misinterpret skewness, sample size, or the presence of measurements exceeding 100.

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