A Researcher Plans A Study In Which A Crucial Step Is Offere

A Researcher Plans A Study In Which A Crucial Step Is Offering Partici

A researcher plans a study in which a crucial step is offering participants a food reward. It is important that the three food rewards be equal in appeal. Thus, a pre-study was designed in which participants were asked which of the rewards they preferred. Of the 60 participants, 16 preferred cupcakes, 26 preferred candy bars, and 18 favored dried apricots. Do these scores suggest that the different foods are differentially preferred by people in general? (Use the .05 significance level.) Use the five steps of hypothesis testing. Sketch the distribution involved. Explain your findings.

Paper For Above instruction

Introduction

The preference for different food rewards in research studies can significantly influence participant engagement and the validity of results. When offering a menu of food rewards, researchers must ensure that these options are equally appealing to avoid bias. A pre-study involving 60 participants assessed preferences among cupcakes, candy bars, and dried apricots. The observed frequencies were 16, 26, and 18, respectively. This paper applies the five steps of hypothesis testing to determine whether these preferences suggest a significant difference in general attractiveness among the foods at the 0.05 significance level.

Step 1: State the Hypotheses

The null hypothesis (H₀) assumes that there is no difference in the preference distribution among the three food items, indicating that all are equally preferred in the population. Formally:

H₀: The population preferences are equal across cupcakes, candy bars, and dried apricots.

The alternative hypothesis (H₁) proposes that preferences are not equal, suggesting at least one food item differs in attractiveness:

H₁: Preferences among the foods are not equal in the population.

Step 2: Specify the Significance Level

The significance level (α) is set at 0.05, meaning that if the probability of observing the data under H₀ (the p-value) is less than 0.05, the null hypothesis will be rejected.

Step 3: Calculate the Test Statistic

The appropriate test here is the Chi-Square goodness-of-fit test. First, calculate the expected frequencies assuming equal preference.

Total participants: N = 60

Expected frequency per food: E = N / 3 = 20

Observed frequencies:

- Cupcakes: O₁ = 16

- Candy Bars: O₂ = 26

- Dried Apricots: O₃ = 18

Chi-square statistic:

χ² = Σ [(O_i - E)² / E]

= (16 - 20)² / 20 + (26 - 20)² / 20 + (18 - 20)² / 20

= (−4)² / 20 + (6)² / 20 + (−2)² / 20

= 16 / 20 + 36 / 20 + 4 / 20

= 0.8 + 1.8 + 0.2

= 2.8

Degrees of freedom (df) = k − 1 = 3 − 1 = 2

Using the chi-square distribution table, the critical value at α = 0.05 and df = 2 is approximately 5.991.

Result:

χ² = 2.8

Interpretation:

The calculated chi-square value is less than the critical value.

Step 4: Make a Decision

Since χ² = 2.8 is less than 5.991, we fail to reject the null hypothesis at the 0.05 significance level.

Step 5: Conclusion and Explanation

The evidence does not support the claim that there is a significant difference in preferences for cupcakes, candy bars, and dried apricots among the population. The observed differences in the sample (16, 26, and 18) could be due to chance variation. Therefore, in the context of the main study, these food options can be considered equally appealing, supporting their use as rewards without concern for bias based on preference.

Sketch of the Distribution

The chi-square distribution with 2 degrees of freedom is skewed to the right, peaking near zero and extending towards higher chi-square values. The critical value of 5.991 marks the boundary for significance at α = 0.05. The calculated value (2.8) falls within the non-rejection region, indicating that the observed distribution is consistent with the expected uniform distribution.

Discussion of Findings

The results suggest that, statistically, there is no significant preference among the three food rewards. This supports the validity of using these items as equally appealing rewards in the subsequent main study. It also implies that participants' preferences in this pre-study sample do not deviate enough from an equal preference distribution to warrant concern. Importantly, these findings are context-specific; different populations or larger samples might yield different outcomes. Nonetheless, this preliminary assessment affirms that, in this context, the reward options are sufficiently balanced in attractiveness.

Additional Considerations

While the chi-square test provides insight into preferences, future research could explore whether subtle individual differences influence preferences and whether larger samples confirm these findings. Moreover, effect size measures, such as Cramer's V, could be calculated to quantify the strength of the deviation from the expected distribution, providing further nuance.

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