Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain."
The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. The pH of rain at one location varies among rainy days according to a Normal distribution with mean 5.43 and standard deviation 0.54.
Exercise 3.31 (in your textbook) concerns the acidity (measured by pH) of rainfall.
A sample of 105 rainwater specimens had mean pH 5.43, standard deviation 0.54, and five-number summary 4.33, 5.05, 5.44, 5.79, 6.81.
Compare the mean and median and also the distances of the two quartiles from the median.
Does it appear that the distribution is quite symmetric? Why?
Choose the most accurate answer below, for these data:
A. The median is relatively resistant measure, while the mean isn't. Therefore the mean gets pulled further away from the direction of a long tail. Since the mean is lower than the median, we conclude that the distribution has a right-hand tail.
B. The third quartile is closer to the median than the first quartile is. This indicates that the third quartile may have been pulled up by a right-hand skew in the distribution.
C. The median is a relatively resistant measure, while the mean isn't. Therefore the mean gets pulled further in the direction of a long tail. Since the mean is smaller than the median, we conclude that the distribution has a left-hand tail.
D. The third quartile is closer to the median than the first quartile is. This indicates that the third quartile may have been pushed down by a left-hand skew in the distribution.
E. The third quartile is closer to the median than the first quartile is. This indicates that the third quartile may have been pushed down by a right-hand skew in the distribution.
F. The mean is a relatively resistant measure, while the median isn't. Therefore the median gets pulled further away from the direction of a long tail. Since the median is lower than the mean, we conclude that the distribution has a right-hand tail.
G. The data on the mean, quartiles and median provide us no indication that this distribution is skewed.
To compare the mean and median and the distances of the two quartiles from the median, we need to understand the properties of these measures and their relationship to skewness.
The mean is calculated by summing all the values in a dataset and dividing by the total number of values. It represents the average value of the dataset. The median, on the other hand, is the middle value when the data is arranged in ascending or descending order. It divides the dataset into two equal halves.
In a symmetric distribution, the mean and median will be approximately equal. However, in skewed distributions, the mean can be pulled away from the median in the direction of the longer tail. This occurs because extreme values have a stronger influence on the mean compared to the median.
Now let's analyze the given answer choices:
A. This choice states that the mean is lower than the median, suggesting a right-hand tail. This aligns with what we would expect in a positively skewed distribution.
B. This choice suggests that the third quartile is closer to the median than the first quartile, indicating a right-hand skew in the distribution. It is plausible because extreme values can pull the upper quartile higher.
C. This choice states that the mean is smaller than the median, implying a left-hand tail. However, this contradicts our understanding of skewness based on the given information.
D. This choice suggests that the third quartile is closer to the median than the first quartile, indicating a left-hand skew in the distribution. This contradicts our understanding of skewness based on the given information.
E. This choice suggests that the third quartile is closer to the median than the first quartile, indicating a right-hand skew. This aligns with what we would expect in a positively skewed distribution.
F. This choice states that the median is lower than the mean, suggesting a right-hand tail. However, this contradicts our understanding of skewness based on the given information.
G. This choice states that the data does not provide any indication of skewness. While this is a possibility, it does not align with our understanding of skewness and the relationship between the mean, median, and quartiles.
Based on the explanations above, the most accurate answer choice is E. The third quartile is closer to the median than the first quartile, indicating that the distribution may have been pushed down by a right-hand skew.
The correct answer is: C. The median is a relatively resistant measure, while the mean isn't. Therefore, the mean gets pulled further in the direction of a long tail. Since the mean is smaller than the median, we conclude that the distribution has a left-hand tail.
Explanation: The fact that the mean is smaller than the median indicates that there is a long tail on the left side of the distribution. This suggests that there are some lower pH values that are pulling the mean towards the left. The median, being a resistant measure, is not as affected by extreme values. Therefore, the distribution is not quite symmetric and has a left-hand tail.