Question

Five pulse rates are randomly selected from a set of measurements. The five pulse rates have a mean of 74.6 beats per minute. Four of the pulse rates are

81, 54, 77, and 68.
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

Part (a)

x = the missing value

The set is {81, 54, 77, 68, x}

Add up those values and divide by 5 to compute the mean. Set this equal to the 74.6 stated. Then solve for x like so:

(81+54+77+68+x)/5 = 74.6

(280+x)/5 = 74.6

280+x = 5*74.6

280+x = 373

x = 373-280

x = 93

Answer: The missing value is 93

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Part (b)

In the previous part, we have n = 5 values and n-1 = 5-1 = 4 of them can be freely selected. The 5th final value is locked in based on what those previous four values are. That 5th value being x mentioned earlier.

This shows that the degrees of freedom is n-1

In other words, one value is determined by the other values in the set.

Answer: n-1

Answer 2

The missing pulse rate is 93 beats per minute. In a list of n values with a known mean, you can freely assign n - 1 values, leaving the last value to be determined by the mean requirement. This gives us n - 1 degrees of freedom.

To find the missing value in a set of five pulse rates with a known mean, we need to set up an equation where the sum of all five pulse rates equals the mean multiplied by five (the number of values).

The mean is given as 74.6 beats per minute and the four known pulse rates are 81, 54, 77, and 68.

Let x represent the missing pulse rate. The equation to solve is:

(81 + 54 + 77 + 68 + x) / 5 = 74.6

Multiplying both sides by 5 gives:

81 + 54 + 77 + 68 + x = 373

Combining the known values gives:

280 + x = 373

Subtracting 280 from both sides to solve for x gives:

x = 373 - 280 = 93

So, the missing pulse rate is 93 beats per minute.

Regarding degrees of freedom, when you are creating a list of n values with a known mean, you can freely assign values to n - 1 of those values. The last value is not free to be assigned because it must be whatever value is required to ensure the mean stays constant.

Therefore, the number of degrees of freedom is n - 1.