Homework 7
Resistance
A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially. Notice that for this data set the standard deviation is not resistant to extreme observations, but the interquartile range is resistant.
Conjecture the shape of the distribution for magnitude. Choose the correct answer below.
The mean is larger than the median, and the distance from Q1 to M is less than the distance from M to Q3, which suggests the distribution of magnitude is skewed right.
Conjecture the shape of the distribution for depth. Choose the correct answer below.
The mean is much larger than the median and is greater than Q3, so the distribution of depth is likely skewed right.
The _______ represents the number of standard deviations an observation is from the mean.
z-score The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. It is found by subtracting the mean from the data value and dividing the result by the standard deviation. The z-score is unitless. It has mean 0 and standard deviation 1.
Explain the circumstances for which the interquartile range is the preferred measure of dispersion. What is an advantage that the standard deviation has over the interquartile range?
The interquartile range, IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the first and third quartiles. The interquartile range is not affected by extreme values. Therefore, when the distribution of data is highly skewed or contains extreme observations, it is best to use the interquartile range as the measure of dispersion because it is resistant. The standard deviation describes how far, on average, each observation is from the mean. It is affected by extreme values, but the advantage that it has over the interquartile range is that it uses all the observations in its computation.
In a certain city, the average 20- to 29-year old man is 69.6 inches tall, with a standard deviation of 3.0 inches, while the average 20- to 29-year old woman is 64.5 inches tall, with a standard deviation of 3.8 inches. Who is relatively taller, a 75-inch man or a 70-inch woman?
The z-score for the man, 1.8, is larger than the z-score for the woman, 1.45, so he is relatively taller.
The accompanying data represent the miles per gallon of a random sample of cars with a three-cylinder, 1.0 liter engine. (a) Compute the z-score corresponding to the individual who obtained 41.5 miles per gallon. Interpret this result. (b) Determine the quartiles. (c) Compute and interpret the interquartile range, IQR. (d) Determine the lower and upper fences. Are there any outliers?
A. Compute the z-score corresponding to the individual who obtained 41.5 miles per gallon. Interpret this result. The z-score corresponding to the individual is . 73 and indicates that the data value is . 73 standard deviation(s) above the mean. B. The first quartile, denoted Q1, is the same as the 25th percentile. Determine the first quartile by first arranging the data in ascending order and finding the median. Then divide the data into two halves; the bottom half will be the observations below (to the left of) the location of the median. The first quartile is the median of the bottom half. C. The interquartile range is the difference between the third and first quartiles D. The lower fence is Q1−1.5(IQR). The upper fence is Q3+1.5(IQR) The lower and upper fences define the boundaries for the outliers. Find all observations that are less than the lower boundary or greater than the upper boundary.
Explain the meaning of the accompanying percentiles. (a) The 10th percentile of the head circumference of males 3 to 5 months of age in a certain city is 41.0. (b) The 90th percentile of the waist circumference of females 2 years of age in a certain city is 52.7 cm. (c) Anthropometry involves the measurement of the human body. One goal of these measurements is to assess how body measurements may be changing over time. The following table represents the standing height of males aged 20 years or older for various age groups in a certain city in 2015. Based on the percentile measurements of the different age groups, what might you conclude?
The kth percentile of a set of data is a value such that k percent of the observations are less than or equal to the value. A. 10% of 3- to 5-month-old males have a head circumference that is 41.0 cm or less. B. 90% of 2-year-old females have a waist circumference that is 52.7 cm or less. C. At each percentile, the heights generally decrease as the age increases. Assuming that an adult male does not grow after age 20, the percentiles imply that adults born in 1990 are generally taller than adults who were born in 1950.