# a=10s Show that a=sqrt((H(1-F)/(F(1-H))In a psychophysical experiment designed to measure performance in a recognitiontask, a subject is presented with a set of pictures of people’s faces. Later,...

a=10s Show that a=sqrt((H(1-F)/(F(1-H))

In a psychophysical experiment designed to measure performance in a recognition

task, a subject is presented with a set of pictures of people’s faces. Later, the

subject is presented with a second set of pictures which contains the previously

shown pictures and some new ones. The subject then is asked to answer “yes” or

“no” to the question “Do you recognize this face?” We would like to determine

a measure of the observer’s ability to discriminate between the previously shown

pictures and the new ones.

If a subject correctly recognizes a face as being one of the previously shown

ones, it is called a “hit.” If a subject incorrectly states that they recognize a face,

when the face is actually a new one, it is called a “false alarm.” The proportion

of responses to previously shown faces which are hits is denoted by H, while the

proportion of responses to new faces which are false alarms is denoted by F.

A measure of the ability of the subject to discriminate between previously shown

faces and new ones is given by

all are in log base 10

s = (1/2)(((Log(H/1-H))-log(F/1-F))

Another measure of sensitivity is given by a=10s. Show that a=sqrt((H(1-F)/(F(1-H))

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### 1 Answer

You need to prove that `a = 10 s` , if `s = (1/2) (log(H/(1-H)) - log(F/(1 - F)))` , hence, using logarithmic identities yields:

`s = (1/2) (log(H/(1-H)) - log(F/(1 - F))) => s = (1/2) log((H/(1-H))/(F/(1 - F)))`

`s = (1/2) (log((H(1 - F))/(F(1 - H))))`

`s = (log((H(1 - F))/(F(1 - H)))^(1/2))`

Converting the power into a square root yields:

`s = log (sqrt((H(1 - F))/(F(1 - H))))`

You need to equate 10 s and a such that:

`a = 10 s => a = 10 log (sqrt((H(1 - F))/(F(1 - H)))) => a = log (((H(1 - F))/(F(1 - H)))^(10/2)) `

`a = log (((H(1 - F))/(F(1 - H)))^5) `

**Hence, evaluating the value of a, under the given conditions, yields that `a = log (((H(1 - F))/(F(1 - H)))^5)` and the indicated result `a = (sqrt((H(1 - F))/(F(1 - H)))) ` does not hold.**