Question

the histogram represents the number of text messages that bea made in one month. use the histogram to answer the following question. which best describes the distribution of the data? bea’s daily text messages over 30 days histogram with x - axis: daily text messages (25, 37, 49, 61, 73, 85, 97) and y - axis: frequency (0 - 10) a skewed right b skewed left c scattered d symmetric e bimodal

Explanation:

Step1: Recall Skewness Definitions

Skewed right: Tail on right (higher values) is longer. Skewed left: Tail on left (lower values) is longer. Symmetric: Both sides mirror. Bimodal: Two peaks. Scattered: No pattern.

Step2: Analyze the Histogram

The left side (lower text message counts: 25) has a short bar, and the right side (higher counts: 97) has a longer tail (bar at 97 is short but the left tail? Wait, no—wait, the bins are 25, 37, 49, 61, 73, 85, 97. The leftmost bin (25) has a small frequency, and the rightmost (97) also small, but the left tail (lower values) is shorter, right tail? Wait, no—wait, skewed left means the tail is on the left (low values). Wait, the histogram: the first bin (25) has frequency ~1, then 37 has 6, 49 has 8, 61 has 7, 73 has 6, 85 has 6, 97 has 2. Wait, the left tail (low values: 25) is short, right tail (high values: 97) is longer? No, wait 25 is left, 97 is right. Wait, the frequency at 25 is 1 (small), then increases to 49 (8), then decreases. Wait, the left side (lower x) has a small bar, right side (higher x) has a small bar but the left tail (low x) is shorter, right tail (high x) is longer? No, skewed left is when the tail is on the left (low x), meaning most data is on the right. Wait, in the histogram, the leftmost bin (25) has low frequency, and as x increases, frequency increases to a peak then decreases, but the left tail (25) is short, right tail (97) is longer? Wait no, let's check skewness: skewed left (negative skew) has mean < median < mode, tail on left. Skewed right (positive skew) has mode < median < mean, tail on right. Looking at the histogram: the first bin (25) has frequency 1 (left tail), then frequencies increase to 49 (8), then decrease. Wait, the left tail (low x) is short, right tail (high x) is longer? Wait no, 25 is left, 97 is right. The bar at 25 is small, then 37 is 6, 49 is 8, 61 is 7, 73 is 6, 85 is 6, 97 is 2. So the left tail (25) is short, right tail (97) is longer? Wait, no—wait, skewed left: tail on left (low values), so the left side (low x) has a long tail. But here, the leftmost bin (25) has a small bar, then the next (37) is higher. So the tail is on the left? Wait, no, the left tail is the part with low frequency on the left (low x). So if the leftmost bin has low frequency, and the next bin has higher, that's a short left tail. Wait, maybe I got it reversed. Let's think: skewed right (positive skew) has the tail on the right (high x), meaning most data is on the left. Skewed left (negative skew) has tail on left (low x), most data on right. Looking at the histogram: the bins from 25 (low) to 97 (high). The frequency at 25 is 1 (low), then 37 is 6, 49 is 8, 61 is 7, 73 is 6, 85 is 6, 97 is 2. So the left tail (25) is short, right tail (97) is longer? Wait, no—97 is right, frequency 2, which is higher than 25's 1. Wait, maybe the distribution is skewed left? Wait, no, let's check the peak. The peak is at 49 (frequency 8). Then, to the left of the peak (25, 37), the frequencies are 1 and 6 (increasing towards peak). To the right of the peak (61, 73, 85, 97), frequencies are 7, 6, 6, 2 (decreasing, then a small increase? No, 61 is 7, 73 is 6, 85 is 6, 97 is 2. So the right side has a longer tail? Wait, no, the left side (before peak) has a small bar (25) then increases, right side (after peak) decreases but has a small bar at 97. Wait, maybe the distribution is skewed left. Wait, the key is: skewed left means the tail is on the left (low values), so the leftmost bins have low frequency, and the data is concentrated on the right. But in this histogram, the leftmost bin (25) has low frequency, then 37 is higher, 49 is peak, then decreases. So the left tail (25) is short, right tail (97) is longer? No, I think I made a mistake. Wait, skewed left: the tail is on the left, so the mean is less than the median, and the left side (low x) has a long tail (low frequency extending to left). But here, the leftmost bin is 25, with frequency 1, then 37 is 6 (higher), so the left tail is short. Skewed right: tail on right (high x), so high x bins have low frequency extending to right. Here, 97 is high x, frequency 2, which is low, and the bins before 97 (85, 73, 61) have higher frequencies. Wait, maybe the distribution is skewed left. Wait, let's count the frequencies: 25:1, 37:6, 49:8, 61:7, 73:6, 85:6, 97:2. The median: there are 1+6+8+7+6+6+2=36 data points (wait, 30 days? Wait the x-axis is 25,37,49,61,73,85,97—maybe the bins are 25-37, 37-49, 49-61, 61-73, 73-85, 85-97? Wait no, the x-axis labels are 25,37,49,61,73,85,97, so each bin is from 25-37, 37-49, etc. So number of days: 1 (25-37), 6 (37-49), 8 (49-61), 7 (61-73), 6 (73-85), 6 (85-97), 2 (97-...)? Wait, maybe the total is 1+6+8+7+6+6+2=36, but the problem says 30 days. Maybe a typo, but regardless, the shape: the left side (low x) has a small bar (25-37:1), then increases to 49-61:8, then decreases, but the right side (high x) has 97-...:2. Wait, the left tail (low x) is short, right tail (high x) is longer? No, 25-37 is 1, 97-... is 2, so right tail is longer. But skewed right is tail on right. Wait, but the peak is at 49-61, then to the left of peak (25-37, 37-49) frequencies are 1 and 6 (increasing), to the right (61-73, 73-85, 85-97, 97-...) frequencies are 7,6,6,2 (decreasing, then a small increase? No, 97-... is 2, which is lower than 85-97's 6. So the right tail is longer (97-... has low frequency). So the distribution is skewed right? Wait no, I'm confused. Wait, let's look at the options: A is skewed right, B skewed left, C scattered, D symmetric, E bimodal. Bimodal would have two peaks, but here only one peak (49-61). Scattered would have no pattern, but here there's a peak. Symmetric would have left and right mirror, but left has 1,6,8 and right has 7,6,6,2—no mirror. Skewed left: tail on left (low x), so left side has low frequency. Here, leftmost bin (25-37) has frequency 1 (low), then 37-49 has 6 (higher), so left tail is short. Skewed right: tail on right (high x), so rightmost bin (97-...) has low frequency. Here, 97-... has 2, which is low, and the bins before (85-97) have 6, 73-85 have 6, 61-73 have 7—so right tail is longer. Wait, but the left side (25-37) is 1, which is a short tail, right side (97-...) is 2, longer? No, 1 is shorter than 2. Wait, maybe the distribution is skewed left. Wait, the key is: in a skewed left distribution, the mean is less than the median, and the tail is on the left (low values). In this histogram, the leftmost bin (low values) has a small frequency, meaning the tail is on the left. Wait, 25-37 is low x, frequency 1 (small), so that's the left tail. Then the data is concentrated on the right (higher x). So skewed left. Yes, that makes sense. Because the left tail (low x) has low frequency, so the tail is on the left, meaning skewed left (B).

Answer:

B. Skewed left