Question

When choosing an item from a group, researchers have shown that an important factor influencing choice is the item's location. This occurs in varied situations such as shelf positions when shopping, filling out a questionnaire, and even when choosing a preferred candidate during a presidential debate. In this experiment, five identical pairs of white socks were displayed by attaching them vertically to a blue background, which was then mounted on an easel for viewing. One hundred participants from the University of Chester were used as subjects and asked to choose their preferred pair of socks.

(a) Suppose each subject selects his or her preferred pair of socks at random. What is the probability that he or she would choose the pair of socks in the center position? (Enter your answer as a proportion rounded to one decimal place.)

?=P=

Assuming the subjects make their choices independently, select the answer choice that correctly identifies the distribution of ?X , the number of subjects among the 100100 who would choose the pair of socks in the center position.

?X has a binomial distribution with parameters ?=100 and ?=15n=100 and p=15

?X has a binomial distribution with parameters ?=20 and ?=15n=20 and p=15

?X has a binomial distribution with parameters ?=4 and ?=1100n=4 and p=1100

?X has a binomial distribution with parameters ?=100 and ?=23n=100 and p=23

(b) What is the mean, ?μ , of the number of subjects who would choose the pair of socks in the center position? (Enter your answer as a whole number.)

?=μ=

subjects

What is the standard deviation, ?σ ? (Enter your answer as a whole number.)

?=σ=

subjects

In choice situations of this type, subjects often exhibit the "center stage effect," which is a tendency to choose the item in the center. In this experiment, 3434 subjects chose the pair of socks in the center. What is the probability, ?P , that 3434 or more subjects would choose the item in the center if each subject were selecting his or her preferred pair of socks at random? Use the Normal approximation first. If your software allows, find the exact binomial probability, ??Pe , and compare the two. (Enter your answers rounded to four decimal places.)

?=P=

??=Pe=

(d) Select the conclusion that correctly identifies whether this experiment supports the "center stage effect".

The experiment does not support the center stage effect. If participants were truly picking the socks at random, it would be highly likely for 3434 or more to choose the center pair.

This experiment does not support the center stage effect because the sample size is too small.

The experiment supports the center stage effect. If participants were truly picking the socks at random, it would be highly unlikely for 3434 or more to choose the center pair.

The experiment is not conclusive. More data is needed.

Answer 1

Date: ----22/04/2019 Answer If each subject selects his or her preferred pair of socks at random, the probability that he or she would choose the pair of socks in the center position out of five identical pairs of white socks-1/5 = 0.2 X has a binomial distribution with parameters n-100 and p-1/5 Mean np 100 (1/5) 20 standard deviation = Vnp(1-P-V 100 * 0.2 (1-02) = 4 standard deviationVnplP Using normal approximation, the probability, P that 34 or more subjects would choose the item in the center if each subject were selecting his or her preferred pair of socks at random P = P(X > 34) P(Z > (34-20) / 4) = P(Z = 3.5) = 0.0002 Using R software, the exact binomial probability, Pe for 34 or more subjects would choose the item in the center is, 0.0003 Both the probabilities, P and Pe are approximately equal. d) As, P or Pe is very low (near zero), if all participants were truly picking the socks at random, it is very unlikely that the 34 or more to choose the center pair. So, the correct answer is

would be highly unlikely for 34 or more to choose the center pair.