Cognitive Science 14
Homework Set 1 Questions

1. A sample was taken of newly divored women older than 40 years. Each gave the age (in nearest years) at which she had been first married. The results were as follows. Calculate the mean, median, and mode of the ages:
   20, 24, 20, 27, 18, 22, 22, 35, 26, 25, 30, 18, 25, 26, 30, 18
   
2. In the same study as problem 1, a sample was taken of married women older than 40 years who had never been divorced. They also reported the age at which they had married, given below. Compute the mean, variance, and standard deviation for both samples. Do the samples differ in the mean age? differ in variability?
   18, 27, 30, 26, 28, 26, 34, 22, 26, 24, 23, 31, 26, 29, 26
   
3. 
   
   

   Find the mean, median, and mode for the distribution shown above. What do the relative sizes of these indices reflect about the distributions form?

4. A study examined the relationship between the rate of cigarettes smoked per day and vital lung capacity. The following results were found. X is the is the smoking rate and Y is the lung capacity:
   

   
   
   

   1. What is the mean smoking rate, ?
   2. What is the variance in smoking rate, ?
   3. What is the mean lung capacity, ?
   4. What is the variance in lung capacity, ?
   5. What is the covariance between rate and capacity, ?

5. Use the information given in Problem 4:
   1. Draw the data in a plot.
   2. What line, , best fits the data?
   3. For each entry in the table, compute the estimated ?
   4. Plot the estimated and draw a line through them.
   5. For each entry in the table, compute the error .
   6. If someone else smoked 8 cigarettes per day, what would we predict to be there lung capacity?
6. Use the information given in Problem 4:
   1. Compute the sum of squared errors, SSE.
   2. Compute the total sum of squares, SST.
   3. Compute the regression sum of squares, SSR.
   4. What percentage of the data is explained by the best fit line (what is the coefficient of determination, )?
   5. What is the correlation, r, between rate and lung capacity?
   6. Is there a linear relationship between smoking rate and lung capacity?
   7. Does this data really prove smoking impairs lung capacity?

7. 

   Above are six x-y plots of data. Answer the questions below by identifying the correct plot labels for each:

   1. Which have a postive linear correlation?
   2. Which one has the strongest linear correlation, largest r?
   3. Which one would have the largest coefficient for its best fit line?
   4. Which ones are not linearly related?
   5. Which are independent (it is impossible to predict y knowing x)?