Cognitive Science 14
Homework Set 3 Questions

1. Suppose that a random has the following probability density function:
   
   
   1. Plot the function.
   2. Is the function f(x) a valid density function? Why?
   In the following questions, assume f(x) is a valid density function:
   1. Determine the probability the variables takes on the value 2?
   2. Determine the probability the takes on a value between -3 and 1?
   3. Determine the probability the variable takes on some value?
   4. Is the variable continuous?
2. A carnival game uses a large wheel that is spun and then stops at a number. Suppose that the wheel is so constructed that it can stop at any o fthe real numbers between 0 and 6, inclusive. Suppose further that each of these numbers has the same likelihood of occuring (that is, the wheel is fair). Then the number where the wheel stops on any trial is the value of a continuous random variable, the a rule given by
   
   Find the following probabilities:
   1. 
   2. 
   3. 
   4. 
3. Using the random variable defined in problem 2:
   1. What is its mean, ?
   2. What is its variance, ?
4. A teacher believes that the class scores of a final exam are normally distributed. If this is correct, what proportion of the class should fall at or below the following z values?
   1. -1.20
   2. 0.96
   3. 1.88
   4. -1.78
   5. -0.43
   6. 2.15
5. A normal distribution has a mean of 500 and a standard deviation of 10. The researcher desires to divide this distribution into five intervals of values such that the probability of each interval will be exactly 0.20. List the desired intervals, starting with the highest.
6. A standardized test was designed so that is should produce a normal distribution for normal adults. The testmake stated that the middle 50 % of the scores for all such adults should lie between 294 and 306. Determine the mean and standard deviation of the distribution.
7. Assume there is a discrete random variable X with the following distribution function
   
   
   1. What is the mean and the variance of this random variable.
   2. Imagine that we did not know the distribution of this random variable. Instead, we only had two samples drawn from the distribution, and . Try to estimate what the real values in the population are for and . Hint: use
      
      and that
      
      
   3. Now assume we had drawn a different sample, and . What would be our estimates this time?
8. Now we will treat and as random variables. Each time we run an experiment and draw two samples, and , we get a different number for and for .
   1. For N=2 samples, what are all the possible values that and can take, and what are their probablities ... i.e. what are the distribution functions?
   2. What is the expected value for ?
   3. What is the expected value for ?
9. Now repeat the previous problem, but use N=3.
   1. Is the expected value of the same as for the different N?
   2. Is the expected value of the same as for the different N?
   3. Is there a better estimate for the population variance than ?
10. A mean from a sample of 36 cases has a value of 100. How probable is a sample mean, , of 100 or more when the population is normal with the following parameters:
    1. Mean 103, standard deviation 10
    2. Mean 80, standard deviation 50
    3. Mean 99, standard deviation 4
11. Before a presidential primary in a certain state, a candidate claimed approximately 45 % of the votes. A newspaper took a random sample of 500 voters and found 37 % would vote for him. If true value is 45 %, how likely is it to get a sample result with 37 % or less? Do you reject the candidates claim that he has 45 %?