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An experiment was conducted 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 preventing 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. In Problem 39 from Section \(10.2,\) we learned that, after watching both the helper and hinderer toy in action, 14 of 16 ten-month-old babies preferred to play with the helper toy when given a choice as to which toy to play with. A second part of this experiment showed the climber approach the helper toy, which is not a surprising action, and then alternatively the climber approached the hinderer toy, which is a surprising action. The amount of time the ten-month-old watched the event was recorded. The mean difference in time spent watching the climber approach the hinderer toy versus watching the climber approach the helper toy was 1.14 seconds with a standard deviation of 1.75 second. Source: J. Kiley Hamlin et al. "Social Evaluation by Preverbal Infants," Nature, Nov. \(2007 .\) (a) State the null and alternative hypothesis to determine if babies tend to look at the hinderer toy longer than the helper toy. (b) Assuming the differences are normally distributed with no outliers, test if the difference in the amount of time the baby will watch the hinderer toy versus the helper toy is greater than 0 at the 0.05 level of significance. (c) What do you think the results of this experiment imply about 10 -month- olds' ability to assess surprising behavior?

Step by step solution

01

State the hypotheses

Formulate the null and alternative hypotheses. Let \(\bar{d} = 1.14\) represent the mean difference in watching times and \text{H}_{0}\ denote the null hypothesis, \text{H}_{a}\ denote the alternative hypothesis.\[\text{H}_{0}: \bar{d} \leq 0 \text{ (Babies do not tend to look at the hinderer toy longer than the helper toy)} \text{H}_{a}: \bar{d} > 0 \text{ (Babies tend to look at the hinderer toy longer than the helper toy)}\]

02

Define the test statistic

To test the hypothesis, use the t-statistic for the difference of means. The formula for the t-statistic in this context is \[ t = \frac{\bar{d} - \mu}{s/\sqrt{n}}, \] where \( \bar{d} = 1.14 \) (mean difference), \( s = 1.75 \) (standard deviation), \( n = 16 \) (sample size), and \( \mu \) is the population mean of the differences.

03

Compute the test statistic

Insert the values into the formula: \[ t = \frac{1.14 - 0}{1.75/\sqrt{16}} = \frac{1.14}{0.4375} = 2.60 \]

04

Determine the critical value

Use the t-distribution table to find the critical value for a one-tailed test at the 0.05 significance level with \( n-1 = 15 \) degrees of freedom. The critical value is approximately 1.753.

05

Compare the test statistic to the critical value

Since \( t = 2.60 \) is greater than the critical value 1.753, reject the null hypothesis.

06

Interpret the results

The test indicates that at the 0.05 level of significance, there is enough evidence to suggest that babies tend to look at the hinderer toy longer than the helper toy.

07

Implications about assessment of surprising behavior

These results imply that ten-month-olds are able to differentiate between expected (helper) and surprising (hinderer) behavior, spending more time observing the surprising behavior of the hinderer toy.

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Key Concepts

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

Null Hypothesis

In the context of infant behavior studies, the null hypothesis represents a skeptical point of view. It assumes that any observed differences in behavior are due to random chance. For our specific exercise, the null hypothesis states:
\(H_0: \bar{d} \leq 0\).
This means that babies do not tend to look at the hinderer toy longer than the helper toy. The null hypothesis serves as the baseline or default assumption that there is no effect or difference. It is essentially stating that the mean difference in watching time between the hinderer and the helper toy is zero or negative.

Alternative Hypothesis

On the other hand, the alternative hypothesis is what we want to test against the null hypothesis. It suggests that there is an effect or a difference. For our study, the alternative hypothesis is:
\(H_a: \bar{d} > 0\).
This indicates that babies tend to look at the hinderer toy longer than the helper toy. The alternative hypothesis is supported if the experimental data shows a significant difference that is unlikely to have occurred by chance.

T-Statistic

The t-statistic is a crucial value in hypothesis testing. It helps determine whether there is enough evidence to reject the null hypothesis. For our experiment, the t-statistic formula is:
\( t = \frac{\bar{d} - \mu}{s/\sqrt{n}} \),
where:

• \( \bar{d} = 1.14 \), the mean difference
• \( s = 1.75 \), the standard deviation
• \( n = 16 \), the sample size

Here, \( \mu \) is the population mean of the differences. Plugging in these values, the computed t-statistic is 2.60. A higher t-value indicates a larger deviation from the null hypothesis, making it stronger evidence against it.

Sample Size

Sample size (denoted as \( n \)) is the number of subjects participated in the study. In this case, the sample size is 16, representing the 16 ten-month-old babies involved in the experiment. Sample size is important because:

• A larger sample size can give more reliable results.
• It reduces the margin of error and increases the confidence in the findings.
• A small sample size, however, might not represent the entire population accurately.

Therefore, a carefully chosen sample size is critical for drawing meaningful conclusions.

Significance Level

The significance level (denoted as \( \alpha \)) represents the threshold for rejecting the null hypothesis. Commonly set at 0.05, it means there is a 5% risk of concluding that there is an effect when there really is not. In our study, the significance level is 0.05.
This value sets the critical region where if the t-statistic falls into, we can reject the null hypothesis. For our experiment, the critical value from a t-distribution table at \( n-1 = 15 \) degrees of freedom for a one-tailed test at the 0.05 level is approximately 1.753. Since our calculated t-statistic of 2.60 is greater than 1.753, we reject the null hypothesis, suggesting that babies indeed tend to look at the hinderer toy longer than the helper toy.