Question

A) The proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.05 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference. 4) Proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.01 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference.

B) Proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.01 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference.

Answer 1

A)

Let the hypotheses be,

Null Hypothesis H0: p = 0.8

Alternative hypothesis H1: p > 0.8

Let n be the number of surveys collected.

Standard error of the proportion = = 0.4 /

Z value for an alpha of 0.05 is 1.64

Critical value to reject H0 = 0.8 + 1.64 * 0.4 / = 0.8 + 0.656 /

Power = 0.8

P(Reject H0 | p = 0.9) = 0.8

P(p > 0.8 + 0.656 / | p = 0.9) = 0.8

=> P[Z > (0.8 + 0.656 / - 0.9)/ 0.4 / ] = 0.8

=> P[Z > (0.656 / - 0.1) / 0.4 / ] = 0.8

=>   (0.656 / - 0.1) / 0.4 / = -0.8416

=> (0.656 / - 0.1) = -0.8416 * 0.4 /

=> (0.656 + 0.33664) / = 0.1

=> = 0.99264 / 0.1

=> n = 9.9264²  = 99 (Rounded to nearest integer)

B)

Z value for an alpha of 0.01 is 2.33

Critical value to reject H0 = 0.8 + 2.33 * 0.4 / = 0.8 + 0.932 /

Power = 0.8

P(Reject H0 | p = 0.9) = 0.8

P(p > 0.8 + 0.932 / | p = 0.9) = 0.8

=> P[Z > (0.8 + 0.932 / - 0.9)/ 0.4 / ] = 0.8

=> P[Z > (0.932 / - 0.1) / 0.4 / ] = 0.8

=>   (0.932 / - 0.1) / 0.4 / = -0.8416

=> (0.932 / - 0.1) = -0.8416 * 0.4 /

=> (0.932 + 0.33664) / = 0.1

=> = 1.26864 / 0.1

=> n = 12.6864²  = 161 (Rounded to nearest integer)