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Can moving their hands help children learn math? This question was investigated in the paper "Gesturing Gives Children New Ideas About Math" (Psychological Science [2009]: \(267-272\) ). Eighty-five children in the third and fourth grades who did not answer any questions correctly on a test with six problems of the form \(3+2+8=-8\) were participants in an experiment. The children were randomly assigned to either a no-gesture group or a gesture group. All the children were given a lesson on how to solve problems of this form using the stratcgy of trying to make both sides of the equation equal. Children in the gesture group were also taught to point to the first two numbers on the left side of the equation with the index and middle linger of one hand and then to point at the blank on the right side ol the equation. This gesture was supposed to emphasize that grouping is involved in solving the problem. The children then practiced udditional problems of this type. All children were then given a test with six problems to solve, and the number of correct answers was recorded for each child. Summary statistics are given below. Is there evidence that learning the gesturing approach to solving problems of this type results in a significantly higher mean number of correct responses? Test the relevant hypotheses using \(\alpha=0.05\).

Step by step solution

01

State the null and alternative hypotheses

Null hypothesis (H0): There is no difference in the mean number of correct responses between the gesture and no-gesture teaching methods. H0: \(\mu_{gesture} - \mu_{no_gesture} = 0\) Alternative hypothesis (H1): There is a significant difference in the mean number of correct responses between the gesture and no-gesture teaching methods. H1: \(\mu_{gesture} - \mu_{no_gesture} \neq 0\)

02

Calculate the test statistic

To compute the test statistic, we use the formula for the t-test: \[t = \frac{(\overline{x}_{gesture} - \overline{x}_{no_gesture}) - 0}{\sqrt{\frac{s^2_{gesture}}{n_{gesture}} + \frac{s^2_{no_gesture}}{n_{no_gesture}}}}\] Here, \(\overline{x}_{gesture}\) and \(\overline{x}_{no_gesture}\) are the sample means of the number of correct responses in the gesture and no-gesture groups, respectively. \(s_{gesture}\) and \(s_{no_gesture}\) are the sample standard deviations of the number of correct responses in the gesture and no-gesture groups, and \(n_{gesture}\) and \(n_{no_gesture}\) are the sample sizes for the gesture and no-gesture groups.

03

Find the critical value

Next, we need to find the critical value (t-critical) for the given significance level of \(\alpha=0.05\). Since we are performing a two-tailed test, we will divide \(\alpha\) by \(2\) and determine the critical value corresponding to \(\alpha/2=0.025\) in a t-distribution table.

04

Compare the test statistic to the critical value

Now, we will compare the calculated test statistic (t) to the critical value (t-critical). If the calculated test statistic is greater than or equal to the critical value, we reject the null hypothesis.

05

Make a conclusion

Based on the comparison between the test statistic and the critical value, we will conclude whether there is a significant difference in the mean number of correct responses between the gesture and no-gesture teaching methods at a significance level of \(\alpha=0.05\). If we reject the null hypothesis, then there is evidence that learning using gestures leads to a significantly higher mean number of correct responses when compared to not using gestures.

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Key Concepts

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

T-test

A t-test is a statistical test used to determine whether there is a significant difference between the means of two groups. It's particularly useful when dealing with small sample sizes. In the context of the exercise, a t-test was applied to compare the mean number of correct answers in the gesture group against the no-gesture group.

To perform a t-test, one needs to calculate a t-statistic using the formula: \[t = \frac{(\overline{x}_{gesture} - \overline{x}_{no\_gesture}) - 0}{\sqrt{\frac{s^2_{gesture}}{n_{gesture}} + \frac{s^2_{no\_gesture}}{n_{no\_gesture}}}}\]Here, \( \overline{x} \) represents the sample mean, \( s \) is the sample standard deviation, and \( n \) is the sample size of each group.

This equation helps in detecting if the observed difference is statistically significant, or if it might have occurred just by chance.

Null Hypothesis

The null hypothesis, symbolized as \( H_0 \), states that there is no effect or no difference.

• In hypothesis testing, the null hypothesis serves as a starting point.
• It is a statement we aim to test and potentially reject.

For this specific study, the null hypothesis was:
"There is no difference in the mean number of correct responses between the gesture and no-gesture groups."
This implies that any observed differences are due to random chance rather than an actual effect of hand gestures on learning.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_1 \), proposes that there is a meaningful difference.
In this case, the alternative hypothesis suggests that the mean number of correct responses in the gesture group is significantly different from the no-gesture group.
The hypothesis can be formulated as:

• \( H_1: \mu_{gesture} - \mu_{no\_gesture} eq 0 \)

The aim of the t-test is to see if there is enough statistical evidence to reject the null hypothesis in favor of the alternative.

Significance Level

The significance level, often represented by \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true.

• A common threshold used is \( \alpha = 0.05 \), implying a 5% risk of a Type I error.
• In simpler terms, it is the threshold for deciding whether a result is statistically significant.

For the task in the exercise, a significance level of 0.05 was selected. This means that if the p-value calculated from the t-test is less than 0.05, the result is considered significant and the null hypothesis can be rejected.

Critical Value

A critical value is a threshold that the test statistic must exceed to reject the null hypothesis.
In a t-test, this value is determined based on the chosen significance level \( \alpha \) and the degrees of freedom \( df \). These are specific to the sample sizes and the test type.

• For a two-tailed test with \( \alpha = 0.05 \), the critical value is found by looking up the t-distribution table for \( \alpha/2 = 0.025 \).
• If the calculated t-statistic is greater than the critical value, the null hypothesis is rejected.

In our scenario, finding and comparing the t-statistic with the critical value helps in determining whether gesturing significantly impacts learning outcomes.