Question

modeling real life the numbers of text messages sent and received in one day by a random sample of 30 teen cellphone users are shown in the table. number of text messages 35 65 64 88 46 42 71 68 65 97 58 47 52 37 84 58 85 46 56 90 78 76 74 36 74 62 65 75 96 72 estimate the population mean \\( \mu \\). round your answer to the nearest tenth. the population mean \\( \mu \\) is about \\( \square \\).

Explanation:

Step1: Sum all data points

First, we add up all the numbers of text messages. Let's list them out: 35, 65, 64, 88, 46, 42, 71, 68, 65, 97, 58, 47, 52, 37, 84, 58, 85, 46, 56, 90, 78, 76, 74, 36, 74, 62, 65, 75, 96, 72.
Now, let's calculate the sum:
35 + 65 = 100; 100 + 64 = 164; 164 + 88 = 252; 252 + 46 = 298; 298 + 42 = 340; 340 + 71 = 411; 411 + 68 = 479; 479 + 65 = 544; 544 + 97 = 641;
641 + 58 = 700 - 1 (wait, 641 + 58 = 699); 699 + 47 = 746; 746 + 52 = 798; 798 + 37 = 835; 835 + 84 = 919; 919 + 58 = 977; 977 + 85 = 1062; 1062 + 46 = 1108; 1108 + 56 = 1164; 1164 + 90 = 1254;
1254 + 78 = 1332; 1332 + 76 = 1408; 1408 + 74 = 1482; 1482 + 36 = 1518; 1518 + 74 = 1592; 1592 + 62 = 1654; 1654 + 65 = 1719; 1719 + 75 = 1794; 1794 + 96 = 1890; 1890 + 72 = 1962. Wait, maybe a better way: let's count the number of data points, which is 30 (since it's a sample of 30 teens). Wait, let's check the number of numbers: the table has 6 rows, 5 columns each, so 6*5=30. Let's re - calculate the sum carefully:

Let's group them:

First row: 35, 65, 64, 88, 46. Sum: 35 + 65 = 100; 64 + 88 = 152; 46. Total: 100+152 + 46=298.

Second row: 42, 71, 68, 65, 97. Sum: 42+71 = 113; 68+65 = 133; 97. Total: 113+133 + 97=343.

Third row: 58, 47, 52, 37, 84. Sum: 58+47 = 105; 52+37 = 89; 84. Total: 105+89 + 84=278.

Fourth row: 58, 85, 46, 56, 90. Sum: 58+85 = 143; 46+56 = 102; 90. Total: 143+102 + 90=335.

Fifth row: 78, 76, 74, 36, 74. Sum: 78+76 = 154; 74+36 = 110; 74. Total: 154+110 + 74=338.

Sixth row: 62, 65, 75, 96, 72. Sum: 62+65 = 127; 75+96 = 171; 72. Total: 127+171 + 72=370.

Now, sum all these row sums: 298 + 343 = 641; 641+278 = 919; 919+335 = 1254; 1254+338 = 1592; 1592+370 = 1962. So the total sum of all 30 data points is 1962.

Step2: Calculate the mean

The formula for the mean (average) is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n$ is the number of data points and $\sum_{i = 1}^{n}x_{i}$ is the sum of the data points. Here, $n = 30$ and $\sum_{i=1}^{30}x_{i}=1962$.

So the mean $\mu\approx\frac{1962}{30}=65.4$. Wait, wait, 1962 divided by 30: 30*65 = 1950, 1962 - 1950 = 12, 12/30 = 0.4, so 65.4? Wait, no, wait, did I calculate the sum correctly? Let's check with another approach. Let's list all numbers and add:

35, 65, 64, 88, 46, 42, 71, 68, 65, 97, 58, 47, 52, 37, 84, 58, 85, 46, 56, 90, 78, 76, 74, 36, 74, 62, 65, 75, 96, 72.

Let's add them one by one:

Start with 0.

0 + 35 = 35

35 + 65 = 100

100 + 64 = 164

164 + 88 = 252

252 + 46 = 298

298 + 42 = 340

340 + 71 = 411

411 + 68 = 479

479 + 65 = 544

544 + 97 = 641

641 + 58 = 699

699 + 47 = 746

746 + 52 = 798

798 + 37 = 835

835 + 84 = 919

919 + 58 = 977

977 + 85 = 1062

1062 + 46 = 1108

1108 + 56 = 1164

1164 + 90 = 1254

1254 + 78 = 1332

1332 + 76 = 1408

1408 + 74 = 1482

1482 + 36 = 1518

1518 + 74 = 1592

1592 + 62 = 1654

1654 + 65 = 1719

1719 + 75 = 1794

1794 + 96 = 1890

1890 + 72 = 1962. Yes, the sum is 1962. Then the mean is 1962 / 30 = 65.4? Wait, no, 30*65 = 1950, 1962 - 1950 = 12, 12/30 = 0.4, so 65.4? Wait, but let's check with a calculator - like approach:

1962 ÷ 30: 30*60 = 1800, 1962 - 1800 = 162, 162 ÷ 30 = 5.4, so 60 + 5.4 = 65.4. Wait, that's correct. Wait, but maybe I made a mistake in the sum? Let's check the number of terms: 30 terms. Let's count:

Row 1: 5 terms (35,65,64,88,46)

Row 2: 5 terms (42,71,68,65,97) - total 10

Row 3: 5 terms (58,47,52,37,84) - total 15

Row 4: 5 terms (58,85,46,56,90) - total 20

Row 5: 5 terms (78,76,74,36,74) - total 25

Row 6: 5 terms (62,65,75,96,72) - total 30. Correct.

So the mean is $\frac{1962}{30}=65.4$. Wait, but let's check again: 30*65 = 1950, 1962 - 1950 = 12, 12/30 = 0.4, so 65.4.

Wait, but maybe I miscalculated the sum. Let's take a few numbers and re - add:

35, 65 (100), 64 (164), 88 (252), 46 (298); 42 (340), 71 (411), 68 (479), 65 (544), 97 (641); 58 (699), 47 (746), 52 (798), 37 (835), 84 (919); 58 (977), 85 (1062), 46 (1108), 56 (1164), 90 (1254); 78 (1332), 76 (1408), 74 (1482), 36 (1518), 74 (1592); 62 (1654), 65 (1719), 75 (1794), 96 (1890), 72 (1962). Yes, sum is 1962. Then mean is 1962/30 = 65.4.

Wait, but let's check with another method. Let's group the numbers by tens:

30 - 39: 35, 37, 36 → 35+37+36 = 108

40 - 49: 46, 42, 47, 46 → 46+42+47+46 = 181

50 - 59: 64? No, 58, 52, 58, 56 → 58+52+58+56 = 224

60 - 69: 65, 64, 68, 65, 62, 65 → 65+64+68+65+62+65 = 65*4 + 64+68+62 = 260 + 194 = 454

70 - 79: 71, 78, 76, 74, 74, 75, 72 → 71+78+76+74+74+75+72 = 71+78=149; 76+74=150; 74+75=149; 72. Total: 149+150=299; 299+149=448; 448+72=520

80 - 89: 88, 84, 85, 90? No, 88, 84, 85 → 88+84+85 = 257

90 - 99: 97, 90, 96 → 97+90+96 = 283

Wait, no, this grouping is wrong. The correct way is to list all numbers:

Wait, maybe my initial calculation is correct. So the mean is 1962/30 = 65.4.

Answer:

65.4