Question

standardized tests for certain subjects, given to high school students, are scored on a scale of 1 to 5. let a represent the score on a randomly selected exam for subject a and let b represent the score on a randomly selected exam for subject b. the distributions of scores for each subjects standardized tests are displayed in the table and the histograms.
which statement correctly compares the shapes of the distributions?
both distributions are roughly unimodal symmetric.
subject as distribution is skewed right and subject bs distribution is skewed left.
subject as distribution is roughly unimodal symmetric and subject bs distribution is skewed left.
subject as distribution is roughly unimodal symmetric and subject bs distribution is skewed right.

Explanation:

Step1: Analyze Subject A's distribution

Looking at the probabilities for subject A: \(P(A)\) values are \(0.18\) (score 1), \(0.20\) (score 2), \(0.20\) (score 3), \(0.21\) (score 4), \(0.15\) (score 5). The distribution is roughly symmetric around the middle - score (score 3). The probabilities on either side of score 3 are relatively balanced. Also, there is a single "peak" (the highest probability is \(0.21\) for score 4, but the values around score 3 are close), so it is roughly unimodal symmetric.

Step2: Analyze Subject B's distribution

For subject B: \(P(B)\) values are \(0.05\) (score 1), \(0.14\) (score 2), \(0.20\) (score 3), \(0.18\) (score 4), \(0.43\) (score 5). The tail of the distribution is on the left - side (lower scores have lower probabilities). A distribution is skewed right if the tail is on the right (higher values have lower probabilities in the tail). Here, since the probability for the higher score (score 5) is much larger, the tail is on the left. A distribution is skewed left if the tail is on the left. But wait, no: a skewed - right distribution has a long tail on the right. A skewed - left distribution has a long tail on the left. In subject B, the probabilities for lower scores (1 and 2) are very low compared to higher scores (especially score 5). So the tail is on the left (because the non - central part of the distribution with lower frequencies is on the left). Wait, no: skewness is about the direction of the tail. If the majority of the data is on the left and the tail is on the right (lower frequencies for higher values), it is skewed right. If the majority of the data is on the right and the tail is on the left (lower frequencies for lower values), it is skewed left. For subject B, since \(P(B = 5)=0.43\) (a large probability for a high score), and \(P(B = 1)=0.05\) and \(P(B = 2)=0.14\) (low probabilities for low scores), the tail is on the left. But wait, no: skewness is defined as follows. If \(mean>median>mode\), the distribution is skewed right (tail on the right). If \(mean < median

Answer:

Subject A’s distribution is roughly unimodal symmetric and subject B’s distribution is skewed left.