Question

Each year about 1500 students take the introductory statistics course at a large university. This year scores on the final exam are distributed with a median of 74 points, a mean of 70 points, and a standard deviation of 10 points. There are no students who scored above 100 (the maximum score attainable on the final) but a few students scored below 20 points. (a) Is the distribution of scores on this final exam symmetric, right skewed, or left skewed? (b) Would you expect most students to have scored above or below 70 points? (c) Can we calculate the probability that a randomly chosen student scored above 75 using the normal distribution? (d) What is the probability that the average score for a random sample of 40 students is above 75? (e) How would cutting the sample size in half affect the standard deviation of the mean?

Answer 1

Answer:

a. Left Skewed

b. More students scored above 70

c. No

d. 0.0008

e. standard deviation increases by 1.4142

Step-by-step explanation:

a. The distribution  is Left skewed. The maximum possible score is 100 and the mean is 70, hence, there is a barrier preventing a long right tail.

-The scores to the left is only 20, the left therefore is longer.

-The median, 74, is higher than the mean, 70, thus the mean is more left leaning.

b.More students will score more than 70.

-the median is 74 suggesting that 50% plus have scored 74points +

- Given the mean is 70 points, it's therefore obvious that more than 50% of the students have score more than 70 points.

c. No. This is not a normal distribution therefore we cannot calculate the probability that a randomly chosen student scored above 75 using this distribution.

d. The sample size  n is large, n>30, so we can use the central limit theorem to approximate the probability:

$Standard \ Error=\frac{\sigma}{\sqrt{n}}\\\\=\frac{10}{\sqrt{40}}\\\\=1.5811\\\\z=\frac{75-70}{1.5811}\\\\=3.1623\\\\\\(PX>70)=P(z>3.1623)\\\\=0.0008$

e. The standard deviation will increase by the square root of the decreament :

$\sqrt{2}=1.4142$

#Standard deviation increases by 1.4142