Answer to Question #177105 in Statistics and Probability for Kim Galvez

Question

III. Answer the following problems comprehensively.

2. A professor grades students on 5 tests, a project, and a final examination. Each test 

counts as 10% of the course grade. The project counts as 20% of the course grade. The 

final examination counts as 30% of the course grade. Samantha has test scores of 70, 

65, 82, 94, and 85. Samantha’s project score is 92. Her final examination score is 80. 

Use the weighted mean formula to find Samantha’s average for the course. Hint: The 

sum of all the weights is 100%.

3. A salesperson records the following daily expenditures during a 10-day trip. ₱185.34 

₱234.55, ₱211.86, ₱147.65, ₱205.60, ₱216.74, ₱1345.75, ₱184.16, ₱320.45, ₱88.12. 

In your opinion, does the mean or the median of the expenditures best represent the 

salesperson’s average daily expenditure? Explain your reasoning.

4. If exactly one number in a set of data is changed, will this necessarily change the mean 

of the set? Explain.

Answer 1

2. Consider the sum of all weights as

10% + 10% + 10% + 10% + 10% + 20% + 30% = 100% = 1

Hence, the total course grade (TCG) is

$TCG = \frac{70 \times 10}{100} + \frac{65 \times 10}{100} + \frac{82 \times 10}{100} + \frac{94 \times 10}{100} + \frac{85 \times 10}{100} + \frac{92 \times 20}{100}+ \frac{80 \times 30}{100} \\ = 7 + 6.5 + 8.2 + 9.4 + 8.5 + 18.4 + 2 \\ = 82$

Hence, the weighted mean (WM) is

$WM = \frac{82}{1} = 82$

So, Samantha’s average for the course is 82.

3.

$Mean = \frac{\sum x}{n} = \frac{185.34 +234.55 +211.86 +147.65 +205.60+216.74 +1345.75 +184.16 +320.45 +88.12}{10} \\ = 314.022$

For Median value, arrange 10 days daily expenditure from smallest to largest

$88.12 $147.65 $184.16 $185.34 $205.60 $211.86 $216.74 $234.55 $320.45 $1345.75

There are 10 value which is even so add 5th and 6th value and divide by 2 to get media:

$Median = \frac{211.86+205.60}{2} \\ = 208.73$

The median of expenditures is best represent the salesperson’s average daily expenditure because median is always best suited for ordinal values.

4. If any number in the data set is changed, then the mean will also be changed. According to the property of mean, mean is dependent on the change of origin and scale. By definition of mean, mean is sum of the observations divided by the number of observations. As you change any number in the data set, then the sum will change, which will result in a change of mean. If the new number added is higher than the old, then the mean will increase and if the older number added is lower than the old, then the mean will decrease.