Question

1. The May 1, 2009 issue of The Montclarian reported the following home sale amounts for a sample of 10 homes in Alameda, CA that were sold the previous month (1000s of $):                                           

                                            350 408 540 555 575 590 608 679 815 1285

1. What would you report as a representative, or typical, value of home sale amount for this period?

2. What prompted your choice in part a?   

3. Suppose the 10th observation had been 985 rather than 1285. How would the median change?

4. Calculate a 10% trimmed mean.

e. Subtract 100 from each observation to obtain a sample of transformed values. What would be the resulting value of the sample variance if the variance for the original data is 67896.28. Answer without actually performing the subtraction.   

f. If the observations were reexpressed in pennies, that is multiply each observation by 100, what would be the resulting value of the sample standard deviation? The standard deviation of the original data is 260.5691. Answer without actually performing the reexpression.                                                                    

Answer 1

• A typical or representative value will be the one which is the central value.
  We choose the central value as the Median.
  
  The Median is the average of the and value of the ordered observation.
  Hence, the median is the average of the and observation.
  
  Hence, the Median is
  
  
• Median is chosen for the above measure as, 1285 is too large a number for the dataset, and may act as an outlier. Hence, this will have an effect on the mean. Hence, median is chosen, as it is least affected by outliers.
  
  
• Even if the observation is changed, the and value does not change. Hence, it won't affect the median value.
  
  
• 10% of 10 observation is 1 observation.
  
  Hence, trimming the lowest 10% of the data, and the highest 10% of the data, we get the following data:
  408, 540, 555, 575, 590, 608, 679, 815
  
  That is, the lowest and the highest value is deleted.
  
  Hence, the 10% trimmed mean is:
  
  
  
• The formula for sample variance is:
  
  
  Now, let,
  
  
  Hence,
  
  
  Hence, new sample variance is:
  
  
  
  
  
  
  Hence, the sample variance remains same on substracting 100 from the data, that is 67896.28.
  
• The formula for sample variance is:
  
  
  Now, let,
  
  
  Hence,
  
  
  Hence, new sample standard deviation is:
  
  
  
  
  
  
  Hence, the new sample standard deviation is 100 ties the old sample standard deviation.
  
  Hence, the new sample standard deviation is: 26056.91.
  
  

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