91Ó°ÊÓ

Preference for the middle? 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. 12 (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? Assuming the subjects make their choices independently, what is the distribution of \(X\), the number of subjects among the 100 who would choose the pair of socks in the center position? (b) What is the mean of the number of subjects who would choose the pair of socks in the center position? What is the standard deviation? (c) In choice situations of this type, subjects often exhibit the "center stage effect, " which is a tendency to choose the ltem in the center. In this experiment, 34 subjects chose the pair of socks in the center. What is the probability 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? Use the Nomal approximation. If your software allows, find the exact binomial probability and compare the two results. (d) Do you feel that thiss experiment supports the "center stage effect"? Explain briefly.

Step by step solution

01

Understanding the scenario

We are given an experiment where 100 subjects are choosing from 5 identical pairs of socks displayed vertically. We need to calculate probabilities under the assumption that each subject picks randomly and independently.

02

Calculate Probability for Choosing Center Socks

Since there are 5 positions for the socks, if a subject picks randomly, the probability of choosing any specific pair, including the one in the center, is calculated as follows: \[ P( ext{choosing center position}) = \frac{1}{5} \]

03

Determine Distribution of X

Let \(X\) represent the number of subjects out of 100 who choose the central socks. Since each chooses independently, \(X\) can be modeled with a Binomial distribution where the probability \(p = \frac{1}{5}\) and number of trials \(n = 100\). Therefore, \(X \sim \text{Binomial}(100, \frac{1}{5})\).

04

Calculate Mean and Standard Deviation

For a Binomial distribution \(X \sim \text{Binomial}(n, p)\), the mean \(\mu = np\) and standard deviation \(\sigma = \sqrt{np(1-p)}\). Substituting \(n = 100\) and \(p = \frac{1}{5}\): \[ \mu = 100 \times \frac{1}{5} = 20 \] \[ \sigma = \sqrt{100 \times \frac{1}{5} \times \frac{4}{5}} = \sqrt{100 \times \frac{4}{25}} = \sqrt{16} = 4 \]

05

Use Normal Approximation for Probability of 34 or more

To approximate the probability of 34 or more subjects selecting the center position, we use the Normal approximation to the Binomial:\(Y \sim N(\mu = 20, \sigma = 4)\). Using continuity correction, consider \(P(X \geq 34)\) as \(P(Y \geq 33.5)\): Standardize using \(Z = \frac{Y - \mu}{\sigma}\): \[ Z = \frac{33.5 - 20}{4} = 3.375 \]We then find \(P(Z \geq 3.375)\) using standard normal distribution tables or software.

06

Compare with Exact Binomial Probability

To find the exact probability, compute \(P(X \geq 34)\) using the Binomial formula or statistical software, and compare it with the approximation from the Normal distribution.

07

Conclude on Center Stage Effect

Compare the observed probability of 34 or more with the expected probability. If the observed probability is significantly higher, it suggests a "center stage effect." Given the calculation shows high deviation from expectation under random choice, the experiment supports the existence of a center stage effect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is at the core of understanding how likely events are to occur. In this particular sock experiment, probability theory helps determine the likelihood of a participant choosing the socks in the central position. Since there are five positions, and each position is equally likely, the probability for any pair being chosen, including the middle pair, is \[P(\text{choosing center}) = \frac{1}{5}.\] This means that for each participant, there's a 20% chance of selecting the central pair if their choice is random.

The experiment assumes that each subject's choice is independent of the others. This makes calculating the overall distribution of choices feasible using probability theory models, like the binomial distribution. Probability theory allows us to model and predict the outcomes in the real world by quantifying uncertainty and risk.

Normal Approximation

When working with large sample sizes, the binomial distribution can be approximated by the normal distribution for simplicity. In the exercise, 100 participants make it suitable to use this approximation.

Here, the normal approximation is used to estimate the likelihood of 34 or more people choosing the center socks. The mean and standard deviation from the binomial distribution \[\mu = 20, \quad \sigma = 4 \] are used as parameters for the normal distribution. The normal curve helps make calculations more manageable, especially with large data sets.

A continuity correction is applied when using this approximation. This involves considering half-units in order to bridge the gap between discrete and continuous data, hence changing the calculation from \[P(X \geq 34) \text{ to } P(Y \geq 33.5).\] This finer adjustment increases the accuracy of the normal approximation for discrete data like our binomial sample.

Statistical Experiment Design

Designing a statistical experiment involves planning how data will be collected and analyzed. This sock experiment is designed to investigate decision-making processes, specifically the 'center stage effect.' The researchers have attempted to eliminate bias by using identical pairs and neutral backgrounds.

Key components in this setup include:

• Subjects: 100 participants, ensuring a relatively large sample size for more reliable insights.
• Random Selection: Assumes each participant chooses independently and randomly, allowing probability models to accurately predict outcomes.
• Position Dwelling: Using a fixed display setup to measure a specific behavioral response, i.e., preference for the center.

This meticulous design helps identify patterns in choice behavior under controlled conditions. By understanding how specific displays affect choice, businesses and marketers can optimize product placements effectively.

Sampling Distribution

The sampling distribution is a probability distribution of a statistic obtained from a large number of samples drawn from a specific population. In our example, the statistic of interest is the number of people choosing the central socks.

Because we assume independent choices, the number of selections for the central pair follows a binomial distribution, characterized by \[X \sim \text{Binomial}(100, \frac{1}{5}).\] This sampling distribution helps calculate the probability of various outcomes when 100 random choices are made.

The mean and standard deviation from this binomial setup inform us about the center of the distribution and its spread. As in the exercise, using the normal approximation gives a bell curve shape around the mean, facilitating easier predictions and understanding of rare events like 34 people choosing the center.

Sampling distributions allow researchers to make inferences about probabilities, helping to conclude whether observed data deviates significantly from what randomness would suggest.