Question

There are 39 employees in a particular division of a company. Their salaries have a mean of $70,000, a median of $55,000, and a standard deviation of $20,000. The largest number on the list is $100,000. By accident, this number is changed to $1,000,000.

a)What is the value of the mean after the change? Write your answer in units of $1000.b)What is the value of the median after the change? Write your answer in units of $1000.c)What is the value of the standard deviation after the change? Write your answer in units of $1000.

Answer 1

a) The new mean after the change is approximately $66,923,000. b) The new median after the change is approximately $60,961,000. c) The new standard deviation after the change is approximately $234,358,000.

Step-by-step explanation:

a) To find the new value of the mean, we need to subtract the original largest number ($100,000) and add the new largest number ($1,000,000) to the sum of the salaries. The sum of the salaries can be calculated by multiplying the mean ($70,000) by the number of employees (39). So, the new sum of the salaries is $(70,000 × 39) - $100,000 + $1,000,000 = $2,610,000. Since there are still 39 employees, the new mean is $2,610,000 divided by 39, which is approximately $66,923. The value of the mean after the change is $66,923,000 (in units of $1000).

b) The median is the middle value in a list of numbers when they are ordered from smallest to largest. After the change, the values in the list would be: $55,000, $55,000, $55,000, ..., $55,000, $66,923, $66,923, ..., $1,000,000. Since there are 39 employees, the middle two values would be $55,000 and $66,923. To find the median, we take the average of these two values: ($55,000 + $66,923) / 2 = $60,961. The value of the median after the change is $60,961,000 (in units of $1000).

c) To find the new standard deviation, we need to recalculate it using the new values. First, we need to find the squared differences between each salary and the new mean. The sum of these squared differences is $(39 × ($66,923 - $66,923)^2) + $1,000,000. Then, we divide this sum by 39 and take the square root to find the new standard deviation. The value of the standard deviation after the change is approximately $234,358,000 (in units of $1000).

Answer 2

a) New mean 78.07692308 thousands of dollars

b) The median does not vary

c) New standard deviation 150.1793799 thousands of dollars

Step-by-step explanation:

We are working in units of $1,000

a)What is the value of the mean after the change?

Let

`s_1,s-2,...,s_38`

be the salaries of the employees that earn less than 100 units.

The mean of the 39 salaries is 55 units so

`\displaystyle(s_1+s_2+...+s_(38)+100)/(39)=55`

and

`s_1+s_2+...+s_(38)+100=55*39=2145`

By accident, the 100 on the left is changed to 1,000

`s_1+s_2+...+s_(38)+100+900=55*39=2145+900\Rightarrow \\\\\Rightarrow s_1+s_2+...+s_(38)+1000=3045`

Dividing by 39 both sides, we get the new mean

`\displaystyle(s_1+s_2+...+s_(38)+1000)/(39)=\displaystyle(3045)/(39)=78.07692308`

b)What is the value of the median after the change?

Since the number of data does not change and only the right end of the range of salaries is changed, the median remains the same; 55

c)What is the value of the standard deviation after the change?

The variance is 400, so

`\displaystyle((s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(100-70)^2)/(39)=400\Rightarrow\\\\\Rightarrow (s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(100-70)^2=400*39=15600`

Adding 864,000 to both sides we get

`(s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(100-70)^2+864000=15600+864000\Rightarrow\\\\\Rightarrow (s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(1000-70)^2=879600`

Dividing by 39 and taking the square root we get the new standard deviation

`\displaystyle((s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(1000-70)^2)/(39)=\displaystyle(879600)/(39)=22553.84615\Rightarrow\\\\\Rightarrow \sqrt{\displaystyle((s_1-70)^2+(s_2-70)^2+...+(s_(38)-70)^2+(1000-70)^2)/(39)}=150.1793799`