Packaging of a children's health food. Refer to the Journal...

Packaging of a children's health food. Refer to the Journal of Consumer Behaviour (Vol. 10,2011 ) study of packaging of a children's health food product, Exercise 8.42 (p. 419 ). Recall that a fictitious brand of a healthy food product-sliced apples-was packaged to appeal to children (a smiling cartoon apple on the front of the package). The researchers compared the appeal of this fictitious brand to a commercially available brand of sliced apples that was not packaged for children. Each of 408 schoolchildren rated both brands on a 5 -point "willingness to eat" scale, with $1={ }^{\text {" }}$ not willing at all" and $5={ }^{\prime *}$ very willing." The fictitious brand had a sample mean score of $3.69,$ while the commercially available brand had a sample mean score of $3.00 .$ The researchers wanted to compare the population mean score for the fictitious brand, $\mu_{F},$ to the population mean score for the commercially available brand, $\mu_{C} .$ They theorized that $\mu_{F}$ will be greater than $\mu_{C}$. a. Specify the null and alternative hypothesis for the test. b. Explain how the researchers should analyze the data and why. c. The researchers reported a test statistic value of 5.71. Interpret this result. Use $\alpha=.05$ to draw your conclusion. d. Find the approximate $p$ -value of the test. e. Could the researchers have tested at $\alpha=.01$ and arrived at the same conclusion?

Packaging of a children's health food. Refer to the Journal of Consumer Behaviour (Vol. 10,2011 ) study of packaging of a children's health food product, Exercise 8.42 (p. 419 ). Recall that a fictitious brand of a healthy food product-sliced apples-was packaged to appeal to children (a smiling cartoon apple on the front of the package). The researchers compared the appeal of this fictitious brand to a commercially available brand of sliced apples that was not packaged for children. Each of 408 schoolchildren rated both brands on a 5 -point "willingness to eat" scale, with $1={ }^{\text {" }}$ not willing at all" and $5={ }^{\prime *}$ very willing." The fictitious brand had a sample mean score of $3.69,$ while the commercially available brand had a sample mean score of $3.00 .$ The researchers wanted to compare the population mean score for the fictitious brand, $\mu_{F},$ to the population mean score for the commercially available brand, $\mu_{C} .$ They theorized that $\mu_{F}$ will be greater than $\mu_{C}$.
a. Specify the null and alternative hypothesis for the test.
b. Explain how the researchers should analyze the data and why.
c. The researchers reported a test statistic value of 5.71. Interpret this result. Use $\alpha=.05$ to draw your conclusion.
d. Find the approximate $p$ -value of the test.
e. Could the researchers have tested at $\alpha=.01$ and arrived at the same conclusion?

Key Concepts

Null and Alternative Hypotheses

In any hypothesis testing framework, defining the null hypothesis (H0) and alternative hypothesis (H1) is fundamental. The null hypothesis typically represents a statement of no effect or no difference, whereas the alternative hypothesis represents the researcher's claim or expectation about a difference. In the context of comparing two means, H0 posits that the population means are equal while H1 indicates a directional difference (one mean is greater than the other).

Paired Samples Design

A paired samples design involves taking measurements from the same subjects under different conditions or treatments. This design is used to eliminate or control for inter-subject variability, allowing a more precise comparison of the two conditions. The subjects' repeated observations are then analyzed to determine if there is a statistically significant difference between the two conditions.

t-Test for Dependent Samples

The t-test for dependent (paired) samples is a statistical procedure used to determine whether the mean difference between paired observations is significantly different from zero. It is especially useful when comparing two related means, such as ratings from the same individuals under two different conditions. This test accounts for the pairing and shared variance between measurements, making it more sensitive than an independent samples test in such contexts.

Test Statistic Interpretation

The test statistic (in this case, a t-value) quantifies the difference between the sample mean differences and what would be expected under the null hypothesis, relative to the standard error of the differences. A larger absolute value of the test statistic provides stronger evidence against the null hypothesis, suggesting that the observed difference is less likely to be due to random chance.

p-Value and Significance Level

The p-value measures the probability of observing a test statistic at least as extreme as the one calculated, assuming that the null hypothesis is true. Comparing the p-value with a pre-determined significance level (such as ? = .05) helps determine whether to reject the null hypothesis. A p-value smaller than the significance level indicates that the results are statistically significant, meaning it is unlikely the observed effect happened by chance. Adjusting the significance level (e.g., to .01) makes the criteria for rejecting the null hypothesis more stringent.

Transcript

Please give Ace some feedback