Question

Five pulse rates are randomly selected from a set of measurements. The five pulse rates have a mean of 65.4 beats per minute. Four of the pulse rates are 57​, 55​, 62​, and 83. a. Find the missing value. b. Suppose that you need to create a list of n values that have a specific known mean. Some of the n values can be freely selected. How many of the n values can be freely assigned before the remaining values are​ determined? (The result is referred to as the number of degrees of​ freedom.)

Answer 1

a) 70

b) n-1

Explanation:

Mean is defined as the sum total of data divided by the total number of data.

`\overline x = (\sum Xi)/(N)`

Xi are individual data

N is the total number of data = 5

Given the mean of the pulse rate = 65.4

a) Let the 5pulse rates be our data as shown: 57​, 55​, 62​, 83 and y where y is the missing value. According to the formula:

`65.4 = (57+55+62+83+y)/(5)`

`65.4 = (257+y)/(5)\\ 257+y = 65.4*5\\257+y = 327\\y = 327-257\\y = 70`

The missing value is 70

b) Since the total list of numbers is n values with a specific known mean, if some of this values can be freely selected, the number of n values that can be freely assigned before the remaining values are​ determined is any values less than n i.e n-1 values.

From the formula for calculating mean:

`\overline x = (\sum Xi)/(N)\\{\sum Xi} = N\overline x`

This shows that the sum of all the values is equal to the product of the total values and the mean value. Since we can freely choose n-1 values, then sum of the set of data can be written as
`\sum \sumx^(n-1) _i__=_1 Xi`

`\sum \sumx^(n-1) _i__=_1 Xi \ =\ N \overline x`

The number of n values which is referred to the degree of freedom is n-1