3.13 The following data give the 2010 gross domestic product (in billions of dollars) for all 50 states. The
data are entered in alphabetical order by state (Source: Bureau of Economic Analysis).
173 49 254 103 1901 258 237 62 748 403
67 55 652 276 143 127 163 219 52 295
379 384 270 97 244 36 90 126 60 487
80 1160 425 35 478 148 174 570 49 164
40 255 1207 115 26 424 340 65 248 39
a. Calculate the mean and median for these data. Are these values of the mean and the median sample
statistics or population parameters? Explain.
b. Do these data have a mode? Explain.
mean: add hem up and divide by 50
median: sort low-to-high, take the average of the two middle values
mode: the one which occurs most often (if any). I see that 49 occurs at least twice. If two or more values occur the same maximum number of times, there is more than one mode.
a. To calculate the mean, we need to sum up all the values and divide by the number of data points.
Sum of all values = 173 + 49 + 254 + 103 + 1901 + 258 + 237 + 62 + 748 + 403 + 67 + 55 + 652 + 276 + 143 + 127 + 163 + 219 + 52 + 295 + 379 + 384 + 270 + 97 + 244 + 36 + 90 + 126 + 60 + 487 + 80 + 1160 + 425 + 35 + 478 + 148 + 174 + 570 + 49 + 164 + 40 + 255 + 1207 + 115 + 26 + 424 + 340 + 65 + 248 + 39 = 15231
Number of data points = 50
Mean = Sum of all values / Number of data points = 15231 / 50 = 304.62
The mean and median are sample statistics because they are calculated based on a subset of data (the 50 states).
To calculate the median, we need to organize the data in ascending order and find the middle value. Since there are 50 data points, the median will be the average of the 25th and 26th values.
Arranging the data in ascending order: 26, 35, 36, 39, 40, 49, 49, 52, 55, 60, 62, 65, 67, 80, 90, 97, 103, 115, 1160, 1207, 127, 143, 148, 163, 164, 174, 1901, 219, 237, 244, 248, 254, 255, 258, 270, 276, 295, 340, 379, 384, 403, 424, 425, 478, 487, 570, 652, 748, 115
Median = (127 + 143) / 2 = 135
b. Yes, these data have a mode. The mode represents the value(s) that occur most frequently in the data set. To identify the mode, we need to find the value(s) with the highest frequency. In this case, we observe that the values 49 and 115 occur twice, which is more frequently than any other value in the set. Therefore, the mode of these data is 49 and 115.
a. To calculate the mean (average) for the given data, you need to sum up all the values and divide by the total number of values. In this case, you would add up all the given GDP values: 173 + 49 + 254 + 103 + 1901 + 258 + 237 + 62 + 748 + 403 + 67 + 55 + 652 + 276 + 143 + 127 + 163 + 219 + 52 + 295 + 379 + 384 + 270 + 97 + 244 + 36 + 90 + 126 + 60 + 487 + 80 + 1160 + 425 + 35 + 478 + 148 + 174 + 570 + 49 + 164 + 40 + 255 + 1207 + 115 + 26 + 424 + 340 + 65 + 248 + 39 = 12444.
Next, divide the sum by the total number of values, which is 50 in this case: 12444 / 50 = 248.88.
So, the mean for these data is approximately 248.88 billion dollars.
The median is the middle value when the data is arranged in ascending or descending order. To find the median, you first need to sort the data from smallest to largest: 26, 35, 36, 39, 40, 49, 49, 52, 55, 60, 62, 65, 67, 80, 90, 97, 103, 115, 1160, 1207, 127, 143, 148, 163, 164, 173, 174, 1901, 219, 237, 244, 248, 254, 255, 258, 270, 276, 295, 340, 379, 384, 403, 424, 425, 478, 487, 570, 652, 748, 115.
Since the number of values is even (50), there is no exact middle value. Instead, you need to find the average of the two middle values, which in this case are the 25th and 26th values. So, the median would be (173 + 174) / 2 = 173.5 billion dollars.
Regarding whether these values are sample statistics or population parameters, we don't have enough information in the question to definitively determine this. In general, when calculating measures from an entire population's data, the results are referred to as population parameters. However, if the data represents a sample from a larger population, the calculated measures would be sample statistics.
b. To determine if the data have a mode, you need to look for the value that appears most frequently in the dataset. From the given data, you can observe that the value 49 appears twice, making it the most frequently occurring value. Therefore, the dataset does have a mode, and the mode is 49 billion dollars.