Fall Semester 2002

You have 50 minutes. Please don't rush. Good luck and do well.

1. Given the following sequence of numbers:

   9, 3, 1, 7, 10

   please calculate the mean and the median.

   The sum of the numbers is 30 so the mean is 6. If you srot the numbers ascending you will find 7 is in the middle, so 7 is the median value.

2. Please calculate the standard deviation for the following sequence of numbers:

   1, 2, 3

   Note: please make it clear how you calculate it.

   The mean is 2, the deviations to the mean are -1, 0, and 1. The sum of these numbers squared is 2. The standard deviation will be the square root of 2 divided by n or n-1 depending on the definition you use.

3. In a normal distribution of scores that has
   1. a mean of 70 and
   2. a standard deviation of 5,

   how many of the scores are lower than 65?

   Please express your answer as a percentage and justify your answer.

   About 50-34 = 16% since 65 is exactly one standard deviation away from the mean.

4. The mean in a normal distribution of scores is 120. We know that about 84% of the scores are below 145. Can you estimate the standard deviation in this normal distribution of scores from the data given? Explain your answer. Please be brief but convincing.

   In a normal distribution 50% of the scores are below the mean and 34% more are between the mean and the mean plus one standard deviation. That's 84%. 84% is exactly what we have below 145, so 145 must be exactly one standard deviation away from the mean. That makes the standard deviation about 145-120=25.

5. Which score is better: a score of 75 in a normal distribution with a mean of 70 and a standard deviation of 5, or a score of 75 in a normal distribution with a mean of 73 and a standard deviation of 2? Please justify your answer.

   They're the same, since in both cases 75 has 84% of the scores in the distribution behind it. One can use z-scores also to obtain the same ordering information.

6. Which score is better: a score of 77.5 in a normal distribution with a mean of 70 and a standard deviation of 5, or a score of 76 in a normal distribution with a mean of 73 and a standard deviation of 2? Please justify your answer.

   The s-zcores are equal so none of the scores is better than the other one.

7. Your task is to find out the probability of getting exactly one head in the toss of four coins. You set up an experiment which produces the following results:

   Number of heads	Frequency of occurrence
   0	273
   1	999
   2	1490
   3	996
   4	241

   Given these results, what's the estimated probability?

   Experimentally 999/(273+999+1490+996+241) which is about 0.25