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15.19 The Gunning Fog index is a measure of reading difficulty based on the average number of words per sentence and the percentage of words with three or more syllables. High values of the Gunning Fog index are associated with difficult reading levels. Independent random samples of six advertisements were taken from three different magazines, and Gunning Fog indices were computed to obtain the data given in the accompanying table ("Readability Levels of Magazine Advertisements"'Journal of Advertising Research [1981]: 45-50). Construct an ANOVA table, and then use a significance level of .01 to test the null hypothesis of no difference between the mean Gunning Fog index levels for advertisements appearing in the three magazines. $$ \begin{array}{lrrrrrr} \text { Scientific American } & 15.75 & 11.55 & 11.16 & 9.92 & 9.23 & 8.20 \\ \text { Fortune } & 12.63 & 11.46 & 10.77 & 9.93 & 9.87 & 9.42 \\ \text { New Yorker } & 9.27 & 8.28 & 8.15 & 6.37 & 6.37 & 5.66 \end{array} $$

Step by step solution

01

Collect & Compute Key Statistics

First, collect all the given values from three magazines and calculate their mean values. \(\bar{X}_{SA} = \frac{15.75 + 11.55 + 11.16 + 9.92 + 9.23 + 8.2}{6} \approx 11.13\), \(\bar{X}_{F} = \frac{12.63 + 11.46 + 10.77 + 9.93 + 9.87 + 9.42}{6} \approx 10.58\) and \(\bar{X}_{NY} = \frac{9.27 + 8.28 + 8.15 + 6.37 + 6.37 + 5.66}{6} \approx 7.35\). Moreover, find the overall mean, \(\bar{X}_{total} = \frac{11.13 + 10.58 + 7.35}{3} \approx 9.68\).

02

Calculate The Sum of Squares

Compute the Sum of Squares Between (SSB) and Sum of Squares Within (SSW). SSB involves the differences of means of each group to the total mean while SSW focuses on the differences within each group. Formulas for these are \(SSB = n_1(\bar{X}_{1} - \bar{X}_{total})^2 + n_2(\bar{X}_{2} - \bar{X}_{total})^2 + n_3(\bar{X}_{3} - \bar{X}_{total})^2\) and \(SSW = \Sigma (X_{i1} - \bar{X}_{1})^2 + \Sigma(X_{i2} - \bar{X}_{2})^2 + \Sigma(X_{i3} - \bar{X}_{3})^2\). Calculate these using the collected means and given samples.

03

Find the Degrees of Freedom and Mean Squares

Find the degrees of freedom, which for Between (dfB) is \(k - 1 = 3 - 1 = 2\) and for Within (dfW) is \(N - k = 18 - 3 = 15\). Later, use these to calculate Mean Square Between (MSB) and Mean Square Within (MSW). These are the ratio of SSB and SSW to their respective degrees of freedom, i.e., \(MSB = SSB/dfB\) and \(MSW = SSW/dfW\).

04

Calculate the F-statistic and Find the P-Value

Calculate the F-statistic by dividing MSB by MSW, i.e., \(F = MSB/MSW\). The F-statistic is the ratio of the variance between groups to the variance within groups. Use this value to find the p-value from an F-distribution table.

05

Decide the Hypothesis Test

Lastly, compare the p-value we got from the F-distribution table with the given significance level (0.01). If the P-value is less than the significance level, reject the null hypothesis; otherwise fail to reject the null hypothesis. That is, if P < 0.01, we conclude that there is a significant difference among the mean Gunning Fog index values of advertisements in the three magazines, else there is no significant difference.

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Key Concepts

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

Gunning Fog Index

The Gunning Fog Index is a popular metric used to determine the reading difficulty of a text. This index measures the complexity by taking into account two main factors: the average number of words per sentence and the percentage of complex words, which are words with three or more syllables.
The formula to calculate the Gunning Fog Index is given by: \[ GFI = 0.4 \times \left( \text{Average Sentence Length} + \text{Percentage of Complex Words} \right) \] High values on the Gunning Fog Index indicate text that is difficult to read, suggesting it is best suited for advanced readers. Conversely, lower values indicate that the text is easier to read and comprehend. This metric helps writers and content creators adjust their language to suit their target audience's reading capability.

Reading Difficulty

The concept of reading difficulty refers to how challenging a text is for a reader to understand. Various factors influence reading difficulty, including vocabulary, sentence length, and structure. More complex vocabulary and longer sentences usually increase difficulty, while simpler language and shorter sentences reduce it.

• **Vocabulary**: Words with three or more syllables are considered complex and can impact a text's readability.
• **Sentence Length**: Longer sentences can be more challenging to follow since they might contain complex ideas and multiple clauses.
• **Text Structure**: The organization and flow of a text also contribute to how readable it is, affecting how ideas are connected and understood.

Understanding reading difficulty is essential in fields such as education and communication, ensuring information is accessible to the intended audience.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to make inferences or decisions about a population based on sample data. It involves comparing a null hypothesis, which asserts no effect or no difference, against an alternative hypothesis, which suggests the presence of an effect or a difference. In the context of ANOVA used in the exercise, we are testing if there is no significant difference between means of groups.
**Key Steps Involved in Hypothesis Testing:**

• **State the Hypotheses**: Define both null and alternative hypotheses. For the exercise, the null hypothesis is that the mean Gunning Fog Index is the same across advertisements from the three magazines.
• **Choose a Significance Level**: Often represented as \(\alpha\), common levels are 0.05 or 0.01. In this case, 0.01 is used.
• **Calculate Test Statistic**: Using collected data to compute a value such as \(F\), which can determine the likelihood that any observed differences are due to random chance.
• **Decision Rule**: Determine whether to reject or not reject the null hypothesis by comparing the p-value to the significance level.

By following these steps, conclusions can be drawn about population parameters.

Multiple Group Comparison

When dealing with multiple groups, as in this exercise with advertisement data from three different magazines, comparing their means effectively can reveal significant insights. ANOVA (Analysis of Variance) is a statistical tool specifically designed for multiple group comparisons.
**Why Use ANOVA?**

**Efficiency**: Instead of performing multiple pairwise tests, ANOVA simultaneously evaluates all group means. This saves time and reduces the likelihood of making a Type I error (false positive).

**Statistical Insight**: It helps to understand if at least one group mean significantly differs from the others, without specifying which one.

**Complex Analyses**: Ideal for experiments or studies involving multiple factors or treatments to assess their effects.

Performing an ANOVA test would involve calculating variances between group means and within group observations and then interpreting the resulting \(F\)-statistic to see if there is a significant difference between groups.