Question

the number of text messages that reza sent each day so far in this billing cycle is shown on the dot plot.
text messages per day
(there is a dot plot with a number line from 1 to 13. the dot plot has dots as follows: 2 at 2, 1 at 3, 3 at 6, 2 at 7, 3 at 8, 1 at 11, 2 at 12)
which statement must be true according to the dot plot?
○ the data is symmetric and shows that he typically sent about 6 to 8 text messages per day.
○ the data is symmetric and shows that he sent 11 or 12 texts with the same frequency that he sent 1 or 2 texts.
○ the data is skewed and shows that he typically sent about 6 to 8 text messages per day.
○ the data is skewed and shows that he sent 11 or 12 texts with the same frequency that he sent 1 or 2 texts.

Explanation:

Step1: Analyze Symmetry/Skewness

Check the dot plot's distribution. The middle (around 6 - 8) has more dots, and the tails (low: 1 - 3; high: 11 - 13) have fewer. So data is skewed? Wait, no—wait, the peak is at 6 - 8, and the left (1 - 3) and right (11 - 13) have less. Wait, actually, the dot plot: let's count dots. At 2: 2 dots, 3:1; 6:3, 7:2, 8:3; 11:1, 12:2. Wait, the center is 6 - 8, with more dots there. The left (1 - 3) and right (11 - 13) have fewer. So is it symmetric? No, because left has 2 + 1 = 3 dots, right has 1 + 2 = 3 dots? Wait, 2 (at 2) +1 (at 3) = 3; 1 (at 11) +2 (at 12) = 3. And 6:3, 7:2, 8:3. Wait, maybe symmetric? Wait, no, the middle is 6 - 8, and the left and right tails have same number of dots. Wait, but the peak is at 6 and 8 (3 dots each), 7 has 2. Wait, maybe the data is symmetric? Wait, no, the left side (1 - 3) has dots at 2 (2), 3 (1); right side (11 - 13) has 11 (1), 12 (2). Wait, 2 and 12: 2 dots vs 2 dots; 3 and 11: 1 dot vs 1 dot. Oh! So the left (2,3) and right (12,11) are mirror images. And 6,7,8: 3,2,3. So the distribution is symmetric around 7? Wait, 6 and 8 both have 3 dots, 7 has 2. So yes, symmetric. Wait, but the options: let's check each.

Option 1: Data symmetric, typically 6 - 8. Yes, because most dots are at 6 - 8. Option 2: Sent 11/12 same as 1/2. 1:0, 2:2; 11:1, 12:2. So 1 has 0, 2 has 2; 11 has 1, 12 has 2. Not same. So option 2 wrong. Option 3: Data skewed? No, because left and right tails are symmetric. Option 4: Data skewed? No. So correct is option 1? Wait, wait, the dot plot: at 1: 0, 2:2, 3:1; 6:3, 7:2, 8:3; 11:1, 12:2, 13:0. So left tail (1 - 3): 0 + 2 +1 = 3; right tail (11 - 13):1 + 2 +0 = 3. Middle (6 - 8): 3 + 2 +3 = 8. So the distribution is symmetric because left and right tails have same number of dots, and middle is symmetric (6 and 8 have 3, 7 has 2). So data is symmetric. And it shows he typically sent 6 - 8 (since most dots are there). So option 1: "The data is symmetric and shows that he typically sent about 6 to 8 text messages per day."

Wait, but let's recheck. The dot plot: x-axis 1 - 13. Dots:

- 2: 2 dots

- 3: 1 dot

- 6: 3 dots

- 7: 2 dots

- 8: 3 dots

- 11: 1 dot

- 12: 2 dots

So mirror image: 2 (2) ↔ 12 (2); 3 (1) ↔ 11 (1); 6 (3) ↔ 8 (3); 7 (2) is middle. So yes, symmetric. So the data is symmetric. Now, "typically sent about 6 to 8"—since 6,7,8 have 3 + 2 + 3 = 8 dots, which is most. So option 1 is correct.

Wait, but earlier I thought maybe skewed, but no—because left and right tails are symmetric. So the correct statement is the first option? Wait, no, wait the options:

First option: "The data is symmetric and shows that he typically sent about 6 to 8 text messages per day."

Second: "The data is symmetric and shows that he sent 11 or 12 texts with the same frequency that he sent 1 or 2 texts." But at 1: 0 dots, 2: 2 dots; 11:1, 12:2. So 1 and 11: 0 vs 1; 2 and 12: 2 vs 2. Not same frequency for 1/2 and 11/12. So second option wrong.

Third: "The data is skewed..." No, because symmetric.

Fourth: "The data is skewed..." No.

So correct is first option? Wait, but let's check the dot plot again. Wait, at 1: 0 dots, 2: 2, 3:1; 6:3, 7:2, 8:3; 11:1, 12:2. So the left (1 - 3) has 0 + 2 +1 = 3 dots; right (11 - 13) has 1 + 2 +0 = 3 dots. Middle (4 - 5): 0 dots. So the distribution is symmetric around 7 (since 6 and 8 are symmetric, 3 and 11, 2 and 12). So data is symmetric. And most dots are at 6 - 8, so "typically sent about 6 to 8" is true. So first option is correct.

Wait, but maybe I made a mistake. Let's re-express:

- Symmetric: left and right mirror. Here, 2 (2 dots) ↔ 12 (2 dots); 3 (1) ↔ 11 (1); 6 (3) ↔ 8 (3); 7 (2) is center. So yes, symmetric.

- "Typically sent about 6 to 8": since 6,7,8 have 3+2+3=8 dots, which is most (total dots: 2+1+3+2+3+1+2= 14? Wait 2 (at 2) +1 (3) +3 (6) +2 (7) +3 (8) +1 (11) +2 (12) = 2+1=3; 3+2+3=8; 1+2=3. Total 14. So 8 out of 14 are at 6 - 8. So yes, typical is 6 - 8.

So the correct option is the first one: "The data is symmetric and shows that he typically sent about 6 to 8 text messages per day."

Answer:

The data is symmetric and shows that he typically sent about 6 to 8 text messages per day. (The first option)