Question

The following is a section of the write-up of your results. Using the information in the output, fill in the blanks with the appropriate words, phrases, or values that correctly describe the results of your study. Use a significance level of α = .05. In the sample, preferred the identical food item in the unmarked wrapper. This difference in preference significant, χ² ( , n = 83) = , p = .

Answer 1

In the sample, 54 preschoolers preferred the identical food item in the unmarked wrapper. This difference in preference is statistically significant, χ² (1, n = 83) = 7.53, p = 0.006.

How is it so?

1. Expected Frequencies: The expected frequency for each group is calculated using the formula: Expected N = (Row Total * Column Total) / Grand Total.

For the marked wrapper:

`\[Expected N_{\text{Marked}} = (29 + 54) * (29 + 41.5)/(83) \\\approx 41.5\]`

For the unmarked wrapper:

`\[Expected N_{\text{Unmarked}} = (29 + 54) * (54 + 41.5)/(83) \\\approx 41.5\]`

2. Residuals: The residual for each group is calculated as the observed frequency minus the expected frequency.

For the marked wrapper:

`\[Residual_{\text{Marked}} = 29 - 41.5 \\= -12.5\]`

For the unmarked wrapper:

`\[Residual_{\text{Unmarked}} = 54 - 41.5 \\= 12.5\]`

3. Chi-Square (χ²) Test Statistic: The chi-square test statistic is calculated using the formula:
`\(\chi^2 = \sum ((O - E)^2)/(E)\),` where O is the observed frequency and E is the expected frequency.

`\[\chi^2 = ((-12.5)^2)/(41.5) + ((12.5)^2)/(41.5) \approx 7.53\]`

4. Degrees of Freedom (df): Degrees of freedom is calculated as the number of categories minus 1. In this case, df = 2 - 1 = 1.

5. P-Value: The p-value is determined from the chi-square distribution table or using statistical software. In this case, the p-value is given as 0.006.

Therefore, the results can be summarized as follows:

`\[ \chi^2(1, n = 83) = 7.53, \text{ p} = 0.006 \]`

The small p-value indicates that the difference in preference for the type of wrapper is statistically significant. The negative and positive residuals indicate the direction of the deviation from expected values for the marked and unmarked wrappers, respectively.

Complete question:

Researchers at Stanford University conducted a taste test with preschoolers. Each preschooler tasted two identical samples of five food items (hamburger, french fries, chicken nuggets, baby carrots, and milk). For each item, the children tasted one sample wrapped in a McDonald’s wrapper and the other sample in an identical, but unbranded, wrapper. Some children tasted the sample in the branded wrapper first, others tasted the sample in the unbranded wrapper first. Consistently for all food items, the majority of preschoolers preferred the food in the McDonald’s wrapper over the one in the unbranded wrapper, even though both samples were identical. (Source: Robinson, T. N., et al (2007). Effects of fast food branding on young children’s taste preferences. Archives of Pediatrics & Adolescent Medicine (161), 792–797).

The researchers found that the preschoolers who watched more television or ate more often at McDonald’s were more likely to prefer the food wrapped in a McDonald’s wrapper. However, is it possible that the children were attracted to the branded wrapper because it had more writing and colors on it, rather than because they recognized the McDonald’s logo?

You decide to conduct a similar experiment among 83 preschoolers using identical samples of carrots. In this experiment, you wrap one of the identical samples in a wrapper that is colorfully marked and lettered with a description of the contents and the other in an unmarked, unlabeled wrapper of the same size, shape, and material.

Suppose in your experiment, 29 of the 83 children preferred the carrots in the wrapper that is colorfully marked and lettered with a description of the contents (the marked wrapper). You use a statistical computing package to conduct a chi-square goodness of fit test. The following tables consist of the output.

Carrots

Observed N

Expected N

Residual

Marked 29 41.5 -12.5

Unmarked 54 41.5 12.5

Total 83

Test Statistics

Carrots

Chi-Square 7.53

df 1

Asymp. Sig 0.006

The following is a section of the write-up of your results. Using the information in the output, fill in the blanks with the appropriate words, phrases, or values that correctly describe the results of your study. Use a significance level of α = .05.

In the sample, preferred the identical food item in the unmarked wrapper. This difference in preference significant, χ² ( , n = 83) = , p = .