Question

a veterinary researcher is studying a particular type of dog called the australian cattle dog. the researcher has acquired data on a sample of 10 dogs. the data for the weight of these dogs in pounds are reproduced in the table below. calculate the mean and median using a ti-83, ti-83 plus, or ti-84 graphing calculator (round your answers to two decimal places). australian cattle dog weights in pounds 36.1 38.7 39.3 39.4 39.9 40.3 40.9 41.9 43 43 provide your answer below: mean = \\(\square\\), median = \\(\square\\), mode = \\(\square\\)

Explanation:

Step1: Calculate the Mean

The mean is the sum of all values divided by the number of values. First, sum the weights: \(36.1 + 38.7 + 39.3 + 39.4 + 39.9 + 40.3 + 40.9 + 41.9 + 43 + 43\). Let's calculate that sum:
\[ \begin{align*} &36.1+38.7 = 74.8\\ &74.8+39.3 = 114.1\\ &114.1+39.4 = 153.5\\ &153.5+39.9 = 193.4\\ &193.4+40.3 = 233.7\\ &233.7+40.9 = 274.6\\ &274.6+41.9 = 316.5\\ &316.5+43 = 359.5\\ &359.5+43 = 402.5 \end{align*} \]
There are 10 values, so the mean is \(\frac{402.5}{10}=40.25\)? Wait, no, wait, let's check the addition again. Wait, 36.1 + 38.7 = 74.8; 74.8 + 39.3 = 114.1; 114.1 + 39.4 = 153.5; 153.5 + 39.9 = 193.4; 193.4 + 40.3 = 233.7; 233.7 + 40.9 = 274.6; 274.6 + 41.9 = 316.5; 316.5 + 43 = 359.5; 359.5 + 43 = 402.5. Wait, but 10 values? Wait the data is 10 values? Wait the table has 10 entries? Let's count: 36.1, 38.7, 39.3, 39.4, 39.9, 40.3, 40.9, 41.9, 43, 43. Yes, 10 values. So sum is 402.5? Wait no, wait 36.1 + 38.7 is 74.8, plus 39.3 is 114.1, plus 39.4 is 153.5, plus 39.9 is 193.4, plus 40.3 is 233.7, plus 40.9 is 274.6, plus 41.9 is 316.5, plus 43 is 359.5, plus 43 is 402.5. So mean is 402.5 / 10 = 40.25? Wait but let's check with a calculator. Wait maybe I made a mistake. Wait 36.1 + 38.7 = 74.8; 74.8 + 39.3 = 114.1; 114.1 + 39.4 = 153.5; 153.5 + 39.9 = 193.4; 193.4 + 40.3 = 233.7; 233.7 + 40.9 = 274.6; 274.6 + 41.9 = 316.5; 316.5 + 43 = 359.5; 359.5 + 43 = 402.5. So mean is 40.25? Wait but let's check the numbers again. Wait 36.1, 38.7, 39.3, 39.4, 39.9, 40.3, 40.9, 41.9, 43, 43. Let's sum them using another method: (36.1 + 43) + (38.7 + 43) + (39.3 + 41.9) + (39.4 + 40.9) + (39.9 + 40.3). Wait 36.1 + 43 = 79.1; 38.7 + 43 = 81.7; 39.3 + 41.9 = 81.2; 39.4 + 40.9 = 80.3; 39.9 + 40.3 = 80.2. Now sum these: 79.1 + 81.7 = 160.8; 160.8 + 81.2 = 242; 242 + 80.3 = 322.3; 322.3 + 80.2 = 402.5. Yes, so sum is 402.5. Number of values is 10, so mean is 402.5 / 10 = 40.25? Wait but wait, maybe I miscounted the number of values. Wait the table has 10 entries? Let's count: 1. 36.1, 2. 38.7, 3. 39.3, 4. 39.4, 5. 39.9, 6. 40.3, 7. 40.9, 8. 41.9, 9. 43, 10. 43. Yes, 10 values. So mean is 40.25? Wait but let's check with a calculator. Wait maybe I made a mistake. Wait 36.1 + 38.7 = 74.8; +39.3 = 114.1; +39.4 = 153.5; +39.9 = 193.4; +40.3 = 233.7; +40.9 = 274.6; +41.9 = 316.5; +43 = 359.5; +43 = 402.5. So 402.5 divided by 10 is 40.25. Wait but let's check the median. The median for 10 values (even number) is the average of the 5th and 6th values when sorted. The data is already sorted: 36.1, 38.7, 39.3, 39.4, 39.9, 40.3, 40.9, 41.9, 43, 43. So 5th value is 39.9, 6th value is 40.3. So median is (39.9 + 40.3)/2 = 40.1. Wait, that's different from my initial thought. Wait 39.9 + 40.3 = 80.2, divided by 2 is 40.1. So median is 40.10. And the mode is the most frequent value, which is 43 (appears twice, others once).

Wait, but let's recalculate the mean. Wait 36.1 + 38.7 = 74.8; +39.3 = 114.1; +39.4 = 153.5; +39.9 = 193.4; +40.3 = 233.7; +40.9 = 274.6; +41.9 = 316.5; +43 = 359.5; +43 = 402.5. 402.5 divided by 10 is 40.25? Wait, but let's check with a calculator. Wait 36.1 + 38.7 = 74.8; 74.8 + 39.3 = 114.1; 114.1 + 39.4 = 153.5; 153.5 + 39.9 = 193.4; 193.4 + 40.3 = 233.7; 233.7 + 40.9 = 274.6; 274.6 + 41.9 = 316.5; 316.5 + 43 = 359.5; 359.5 + 43 = 402.5. So 402.5 / 10 = 40.25. So mean is 40.25, median is 40.10, mode is 43.

Wait, but let's confirm with the TI-83 steps. To calculate mean and median on TI-83:

1. Enter the data into a list: Press STAT, then EDIT, then enter the data into L1.

2. Press STAT, then CALC, then 1-Var Stats, then enter L1 (if not already), then press ENTER.

The output will give \(\bar{x}\) (mean) and Med (median). Let's simulate that. The data is [36.1, 38.7, 39.3, 39.4, 39.9, 40.3, 40.9, 41.9, 43, 43].

Calculating mean: sum is 36.1 + 38.7 + 39.3 + 39.4 + 39.9 + 40.3 + 40.9 + 41.9 + 43 + 43. Let's add them again:

36.1 + 38.7 = 74.8

74.8 + 39.3 = 114.1

114.1 + 39.4 = 153.5

153.5 + 39.9 = 193.4

193.4 + 40.3 = 233.7

233.7 + 40.9 = 274.6

274.6 + 41.9 = 316.5

316.5 + 43 = 359.5

359.5 + 43 = 402.5

Number of data points (n) = 10

Mean (\(\bar{x}\)) = 402.5 / 10 = 40.25

Median: For n = 10 (even), median is average of (n/2)th and (n/2 + 1)th term. n/2 = 5, n/2 + 1 = 6.

5th term: 39.9

6th term: 40.3

Median = (39.9 + 40.3) / 2 = 40.10

Mode: The value that appears most frequently. 43 appears twice, others once. So mode is 43.

Answer:

Mean = 40.25, Median = 40.10, Mode = 43