Statistics Chapter 3 Homework Questions and Answers 100% Pass

3.1 A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will likely be larger, the mean or the median? Why? Choose the correct answer below. A. The mean will likely be larger because the extreme values in the left tail tend to pull the mean in the opposite direction of the tail. B. The median will likely be larger because the extreme values in the right tail tend to pull the median in the direction of the tail. C. The mean will likely be larger because the extreme values in the right tail tend to pull the mean in the direction of the tail. D. The median will likely be larger because the extreme values in the left tail tend to pull the median in the opposite direction of the tail. - Answer-C. The mean will likely be larger because the extreme values in the right tail tend to pull the mean in the direction of the tail. Note: When data are either skewed left or skewed right, there are extreme values in the tail, which tend to pull the mean in the direction of the tail. If the distribution of the data is skewed right, there are large observations in the right tail. These observations tend to increase the value of the mean, while having little effect on the median. 3.1 True or False: A data set will always have exactly one mode. Choose the correct answer below. a. False b. True - Answer-a. False Note: The mode of a variable is the most frequent observation of the variable that occurs in the data set. To compute the mode, tally the number of observations that occur for each data value. The data value that occurs most often is the mode. A set of data can have no mode, one mode, or more than one mode. If no observation occurs more than once, the data have no mode. 3.1 Example Find the population mean or sample mean as indicated. Sample: 14, 10, 11, 4, 21 - Answer-The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set, divided by the number of observations. The population arithmetic mean, μ, is computed using all the individuals in a population. The sample arithmetic mean, x, is computed using sample data. For this problem, you are finding x because the data set is a sample. To find the sample mean, x, start by calculating the sum of the sample. 14+10+11+4+21=60 Now determine the size of the sample, n. There are five values in the data set. Therefore, n=5. Finally, to calculate the mean of the sample divide the sum by the size of the sample. x = 60 / 5 = 12 3.1 Find the population mean or sample mean as indicated. Sample: 20, 15, 2, 13, 25 Select the correct choice below and fill in the answer box to complete your choice. A. x=___ B. μ=____ - Answer-A. x= 15 Note: Added all together then divided by 5 since there are 5 numbers 3.1 Example The following data represent the miles per gallon for a particular make and model car for six randomly selected vehicles. Compute the mean, median, and mode miles per gallon. 41.1, 38.4, 21.9, 32.5, 28.6, 30.7 - Answer-The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. The mode of a variable is the most frequent observation of the variable that occurs in the data set. The formula for x, the arithmetic mean, is shown below, where x1, x2, ..., xn are n observations of a variable from a sample. x= (x1+x2+•••+xn) / n = (∑ x i) / n While either technology or the formula can be used to find the mean, in this problem, use technology. Compute the mean. x=32.2 Thus, the mean mileage per gallon for the six cars is 32.2. To find the median of a data set, first arrange the data in ascending order, then determine the number of observations, n, and, finally, determine the observation in the middle of the data set. If the number of observations is odd, then the median is the data value that is exactly in the middle of the data set. That is, the median is the observation that lies in the (n+1)/2 position. If the number of observations is even, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observations that lie in the n/2 position and the (n/2)+1 position. While either technology or the process described above can be used to find the median, in this problem, use technology. Determine the median. M=31.6 Thus, the median mileage per gallon for the 6 cars is 31.6. To determine the mode, start by determining the frequency of each mileage per gallon. If no observation occurs more than once, the data are said to have no mode. Note that no value has a frequency other than 1. Since none of the values are The following data represent the miles per gallon for a particular make and model car for six randomly selected vehicles. Compute the mean, median, and mode miles per gallon. 22.6, 25.1, 31.7, 27.7, 22.9, 39.6 1. Compute the mean miles per gallon. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mean mileage per gallon is_______ (Round to two decimal places as needed.) B. The mean does not exist. 2. Compute the median miles per gallon. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The median mileage per gallon is ________ (Round to two decimal places as needed.) B. The median does not exist. 3. Compute the mode miles per gallon. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mode is _____ - Answer-1. a. The mean mileage per gallon is 28.27 2. a. The median mileage per gallon is 26.4 3. B. no mode 3.1 Example A concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The following data represent the strength of nine randomly selected casts (in psi). 3975, 4085, 3100, 3200, 2950, 3840, 4085, 4020, 3415 Compute the mean, median and mode strength of the concrete (in psi). - Answer-The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set, divided by the number of observations. In order to calculate the mean start by summing the strengths of the concrete. 3975+4085+3100+3200+2950+3840+4085+4020+3415=32,670 Now determine the number of observations, n, in the sample. There are n=9 observations. Find the mean by dividing the sum by the number of observations. x = 32,670 / 9 = 3630 psi Thus, the mean strength of the concrete is 3630 psi of pressure. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. That is, half the data are below the median and half the data are above the median. We use M to represent the median. To find the median start by arranging the data in ascending order. 2950, 3100, 3200, 3415, 3840, 3975, 4020, 4085, 4085 Since the data set has an odd number of values, the median is the data value that is exactly in the middle of the data set. M=3840 psi Thus, the median strength of the concrete is 3840 psi of pressure. The mode of a variable is the most frequent observation of the variable that occurs in the data set. To determine the mode, start by determining the frequency of each strength. Since 4085 psi occurs more than once in the data set, it has a frequenc