An Organizational Psychologist was interested in conducting a study, to determin

Question

An Organizational Psychologist was interested in conducting a study, to determine whether upgrading her company's older computer systems to newly released, faster machines would cause an increase in productivity. Currently, the average level of productivity for the company was 120 with a standard deviation of 20. For this study, she plans to test a new system in a single department where she will be able to evaluate the productivity of 45 people. She expects that the new systems will increase productivity by 10 points. a.) if the 5% significance level were used, what would be the power of her planned study? (Be sure to show all work) b.) what effect size should she obtain? c.) should she plan to use more than 45 participants in his sample? why or why not?

Explanation / Answer

Let known productivity average is 120 with standard deviation 20,

We want to test the hypothesis that
H0: µ = 120 vs H1: µ = 130

let level of significance = 0.05, hence Z/2 = 1.96

a) We have to find power of the study
Let, P( Z < Z/2 | µ = 130 ) = 0.95
i.e. P( (Xbar - µ0) * sqrt(n) / | µ = 130 ) = 0.95

(Xbar - 120) * sqrt(45) / 20 = 1.96,

Hence solving for Xbar, we will get Xbar = 125.8436,

Hence Power is given by P( Xbar < 125.8436 |  µ = 130 )

= P( (Xbar - 130) * sqrt(45) /20 < (125.8436 - 130) * sqrt(45) /20 )

= P( Z < -1.3941 ) = 0.081643

b) The sample size would be effective when power is more in the sense power should be at least 0.80 ,

Now we have to revert above part, i.e. we have power = P(Z < z ) = 0.80,

so from standard normal probability table we can find z = 0.841621, where = 20 and µ = 130,

So reverting the formula  (125.8436 - 130) * sqrt(n) /20 = 0.84621

hence solving for n we get sqrt(n) = 0.84621 * 20 / ( 125.8436 - 130 ) = -4.0976
Hence squaring both the sides we get n = 16.40056 i.e. n = 17 ( by approximating to next integer)

c) She should not to plan to use more than 45 participants in his sample because her power of study should goes on decreasing as she increase the sample size, i.e. the probability of accepting the null hypothesis while it is false will increase as the sample size increase.

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