91Ó°ÊÓ

Yale University graduate student J. Kiley Hamlin conducted an experiment in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed to choose which toy he or she wished to play with. In 14 of the 16 cases, the baby chose the helper toy. Source: J. Kiley Hamlin et al., "Social Evaluation by Preverbal Infants." Nature, Nov. 2007. (a) Why is it important to randomly expose the baby to the helper or hinderer toy first? (b) What would be the appropriate null and alternative hypotheses if the researcher is attempting to show that babies prefer helpers over hinderers? (c) Use the binomial probability formula to determine the \(P\) -value for this test. (d) In testing 12 six-month-old babies, all 12 preferred the helper toy. The \(P\) -value was reported as \(0.0002 .\) Interpret this result.

Step by step solution

01

Random Exposure Importance

Randomly exposing babies to either the helper or hinderer toy first is important to ensure that there is no bias in the order of presentation. This controls for any influence that the sequence of exposure might have on the baby's choice, helping to isolate the effect of the toy's behavior as the variable being measured.

02

Null and Alternative Hypotheses

Formulate the hypotheses for the experiment: - Null hypothesis (H_0): Babies do not show a preference for the helper toy; they choose the toys randomly (i.e., probability of choosing the helper toy is 0.5).- Alternative hypothesis (H_A): Babies prefer the helper toy over the hinderer toy (i.e., probability of choosing the helper toy is greater than 0.5).

03

Calculate P-value

Use the binomial probability formula to calculate the P-value.The occurrence of choosing the helper toy can be modeled by a binomial distribution with parameters \( n = 16 \) and \( p = 0.5 \). The P-value is the probability of observing 14 or more babies choosing the helper toy by chance.Use the binomial cumulative distribution function (CDF) to find this probability: \[ P(X \geq 14) = \sum_{k=14}^{16} \binom{16}{k} p^k (1-p)^{16-k} \]\[ = \binom{16}{14} (0.5)^{16} + \binom{16}{15} (0.5)^{16} + \binom{16}{16} (0.5)^{16} \]\[ = 120 \cdot (0.5)^{16} + 16 \cdot (0.5)^{16} + 1 \cdot (0.5)^{16} \]\[ = 137 \cdot (0.5)^{16} \]\[ \approx 0.0107 \]So, the P-value for 14 out of 16 babies choosing the helper toy is approximately 0.0107.

04

Interpret P-value for Six-Month-Old Babies

For an experiment with 12 six-month-old babies where all 12 preferred the helper toy, a P-value of 0.0002 is given.This P-value is much smaller than common significance levels like 0.05 or 0.01. A P-value of 0.0002 indicates there is a very low probability of observing all 12 babies choosing the helper toy by random chance if there were no true preference. This suggests strong evidence against the null hypothesis, supporting the conclusion that six-month-old babies do indeed prefer the helper toy.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Random Exposure

Random exposure in an experiment like the baby-preference study is crucial. It helps to ensure that any observed effects are due to the variable being tested (in this case, the toy's behavior) and not the order of presentation. By randomly showing either the helper or hinderer toy first, researchers can control for any biases that might arise if one toy were always shown before the other. This randomness helps isolate the effect of the helper or hinderer behavior on the baby's choice. Without random exposure, the sequence could influence the results, leading to incorrect conclusions.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a default statement that there is no effect or no difference in the context being studied. In the baby-preference experiment, the null hypothesis would be that babies do not show a preference for the helper toy; they choose between the helper and hinderer toys randomly. Mathematically, this means the probability of choosing the helper toy is 0.5. This hypothesis sets a baseline that any deviation from this 50-50 choice must be statistically significant to suggest a genuine preference.

Alternative Hypothesis

In hypothesis testing, the alternative hypothesis \(H_A\) suggests that there is an effect or difference. For the baby-preference study, the alternative hypothesis asserts that babies do have a preference for the helper toy over the hinderer toy. Mathematically, this means the probability of choosing the helper toy is greater than 0.5. The alternative hypothesis is what researchers aim to support; showing that babies prefer the helper toy indicates a meaningful deviation from random choice.

Binomial Probability

The binomial probability formula helps calculate the likelihood of a given number of successes in a fixed number of trials, given a specified probability of success. In this case, the 'success' is a baby choosing the helper toy. The formula is expressed as: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( \binom{n}{k} \) is a binomial coefficient, \(p\) is the probability of choosing the helper toy, and \( n \) is the total number of trials (babies). For the experiment, we use this formula to find the probability of at least 14 out of 16 babies choosing the helper toy, summing the probabilities for 14, 15, and 16 babies.

P-value Interpretation

The P-value helps determine the significance of the results in hypothesis testing. It measures the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. A low P-value suggests that such extreme results are unlikely under the null hypothesis, thus providing evidence against \(H_0\). In the baby-preference study, if the P-value is significantly low (e.g., 0.0107 for 14 out of 16) compared to common significance levels like 0.05, it suggests a strong preference for the helper toy. A very low P-value like 0.0002 (in the six-month-old baby test) indicates even stronger evidence against the null hypothesis, reinforcing the conclusion that babies prefer the helper toy.