91Ó°ÊÓ

A group of 115 University of Iowa students was randomly divided into a build- up condition group \((m=56)\) and a scale-down condition group \((n=59)\). The task for each subject was to build his or her own pizza from a menu of 12 ingredients. The build-up group was told that a basic cheese pizza costs \(\$ 5\) and that each extra ingredient would \(\operatorname{cost} 50\) cents. The scale-down group was told that a pizza with all 12 ingredients (ugh!!!) would cost \(\$ 11\) and that deleting an ingredient would save 50 cents. The article "A Tale of Two Pizzas: Building Up from a Basic Product Versus Scaling Down from a Fully Loaded Product" (Market. Lett., 2002: 335-344) reported that the mean number of ingredients selected by the scale-down group was significantly greater than the mean number for the build-up group: \(5.29\) versus \(2.71\). The calculated value of the appropriate \(t\) statistic was \(6.07\). Would you reject the null hypothesis of equality in favor of inequality at a significance level of \(.05\) ? \(.01 ? .001 ?\) Can you think of other products aside from pizza where one could build up or scale down? [Note: A separate experiment involved students from the University of Rome, but details were a bit different because there are typically not so many ingredient choices in Italy.]

Step by step solution

01

Formulate the Hypotheses

The null hypothesis \(H_0\) states that there is no difference in the mean number of ingredients selected between the build-up group and the scale-down group: \(H_0: \mu_1 = \mu_2\). The alternative hypothesis \(H_a\) indicates a significant difference: \(H_a: \mu_1 eq \mu_2\).

02

Identify the Given Information

We know the calculated \(t\) statistic is 6.07. The sample sizes are \(m=56\) and \(n=59\). The significance levels to test are \(\alpha = 0.05\), \(0.01\), and \(0.001\).

03

Determine the Critical Values for the t-test

Since the sample sizes are slightly different, you can approximate degrees of freedom using \(df = m+n-2 = 115\). Look up the critical \(t\) values for \(df = 113\) (approximating for simplicity) at different significance levels: \(\alpha = 0.05\), \(0.01\), \(0.001\). They are approximately 1.98, 2.63, and 3.37, respectively.

04

Compare the Calculated t Value with Critical Values

The calculated \(t\) value is 6.07. Since 6.07 is greater than 1.98 (\(\alpha = 0.05\)), 2.63 (\(\alpha = 0.01\)), and 3.37 (\(\alpha = 0.001\)), we reject the null hypothesis at all tested significance levels.

05

Conclusion

Since the calculated \(t\) value significantly exceeds the critical values at all significance levels tested, we reject the null hypothesis. This indicates a statistically significant difference in the mean number of ingredients chosen by the build-up and scale-down groups.

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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 fundamental statistical tool used to determine if there is a significant difference between the means of two groups. In this exercise, we use the t-test to compare the mean number of pizza ingredients selected by two different groups: the build-up condition group and the scale-down condition group. The t-test helps us determine whether any observed difference in means is due to random chance or a genuine effect.

Here's a simple idea of how the t-test works:

• First, it calculates the difference between the two sample means.
• Then, it considers the variability or deviation within each group.
• Finally, it uses both elements to determine how likely it is that the observed difference between the groups occurred by chance.

In the given exercise, the calculated t-value is 6.07, which is quite large, suggesting a meaningful difference between the two groups' means.

null hypothesis

The null hypothesis, often denoted as \(H_0\), is a fundamental concept in hypothesis testing. It assumes that there is no effect or no difference in the situation being studied. In the pizza study, the null hypothesis stated that the mean number of ingredients chosen by the build-up and scale-down groups was the same. This can be formally written as \(H_0: \mu_1 = \mu_2\).

The purpose of testing the null hypothesis is to see if there's enough evidence to prove it wrong, showing that there is indeed a difference or effect:

• If the data provides strong evidence against \(H_0\), we reject it.
• A failure to reject \(H_0\) implies insufficient evidence against the assumption of no difference.

In this exercise, observed data led to rejecting the null hypothesis, implying a significant difference in the ingredient choices of the two groups.

statistical significance

Statistical significance is a term that describes whether the result of an experiment is likely due to something other than random chance. In the context of a hypothesis test, like our t-test, statistical significance indicates that the result is unlikely to have occurred under the assumption of the null hypothesis.

When we perform hypothesis tests:

• We choose a significance level, denoted by \(\alpha\), which defines how extreme results must be to reject the null hypothesis. Common levels include 0.05, 0.01, and 0.001.
• In this exercise, the critical t-values for these levels were 1.98 (0.05), 2.63 (0.01), and 3.37 (0.001).
• Our calculated t-value of 6.07 surpassed all these critical values.

Hence, the result was statistically significant, allowing us to confidently reject the null hypothesis at all tested significance levels.

degrees of freedom

Degrees of freedom is a statistical concept that refers to the number of independent values that can vary in an analysis without breaking any constraints. For a t-test, degrees of freedom are essential because they help to determine the critical t-values needed for hypothesis testing.

In our pizza experiment:

• Degrees of freedom were calculated as the total number of observations minus the number of groups being compared. For two groups, it’s computed as \(m + n - 2\).
• With 56 and 59 participants in each group, our degrees of freedom were 115.
• This information allowed us to look up the appropriate critical t-values to determine statistical significance accurately.

Degrees of freedom give us the context needed to interpret our test statistics correctly, ensuring we make valid conclusions from our data.