A random sample of 250 undergraduate students from a large university was found to contain 57 freshmen, 62 sophomores, 59 juniors, and 72 seniors. (a) What proportion of undergraduate students are freshmen? (b) What is the risk of being a sophomore?

a) 0.26

b) 0.264

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 250

Number of freshmen = 65

Number of sophomores = 66

Number of juniors = 61

Number of seniors = 58

(a) What proportion of undergraduate students are freshmen

(b) What is the risk of being a sophomore

The screen shots are for A/B :)

So for this case we have a higher probability for the red exam so we can conclude that the student is more likely to score below 20% on the difficult questions in the red one.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the scores for the blue exam, and for this case we know the distribution for X is given by:

Where and

We are interested in the probability that P(X<20%)

And the best way to solve this problem is using the normal standard distribution and the z score given by:

And if we replace we got:

Let Y the random variable that represent the scores for the red exam, and for this case we know the distribution for X is given by:

Where and

We are interested in the probability that P(Y<20%)

And the best way to solve this problem is using the normal standard distribution and the z score given by:

And if we replace we got:

So for this case we have a higher probability for the red exam so we can conclude that the student is more likely to score below 20% on the difficult questions in the red one.