91Ó°ÊÓ

Many people now turn to the Internet to get information on health-related topics. The paper "An Examination of Health, Medical and Nutritional Information on the Internet: A Comparative study of Wikipedia, WebMD and the Mayo Clinic Websites" (The International Journal of Communication and Health [2015]: 30-38) used Flesch reading ease scores (a measure of reading difficulty based on factors such as sentence length and number of syllables in the words used) to score pages on Wikipedia and on WebMD. Higher Flesch scores correspond to more difficult reading levels. The paper reported that for a representative sample of health-related pages on Wikipedia, the mean Flesch score was 26.7 and the standard deviation of the Flesch scores was \(14.1 .\) For a representative sample of pages from WebMD, the mean score was 43.9 and the standard deviation was 19.4 . Suppose that these means and standard deviations were based on samples of 40 pages from each site. Is there convincing evidence that the mean reading level for health-related pages differs for Wikipedia and WebMD? Test the relevant hypotheses using a significance level of \(\alpha=0.05\)

Step by step solution

01

State the null and alternative hypotheses

We want to test if there is a difference in the mean reading level for health-related pages on Wikipedia and WebMD. Let \(\mu_{w}\) be the population mean Flesch score for Wikipedia and \(\mu_{m}\) be the population mean Flesch score for WebMD. The hypotheses are: Null hypothesis (H0): There is no difference in the mean reading levels. \(\mu_{w} - \mu_{m} = 0\). Alternative hypothesis (H1): There is a difference in the mean reading levels. \(\mu_{w} - \mu_{m} \neq 0\).

02

Identify the test statistic and its distribution

We will use a two-sample t-test to compare the mean Flesch scores for Wikipedia and WebMD. The test statistic is given by: \[t = \frac{(\bar{x}_{w} - \bar{x}_{m}) - \Delta}{\sqrt{\frac{s^2_{w}}{n_{w}} + \frac{s^2_{m}}{n_{m}}}}\] where \(\bar{x}_{w}\) and \(\bar{x}_{m}\) are the sample means, \(s^2_{w}\) and \(s^2_{m}\) are the sample variances, \(n_{w}\) and \(n_{m}\) are the sample sizes, and \(\Delta\) represents the difference in population means under the null hypothesis, which is 0 in this case. Since the populations are assumed to be independent, the t-test statistic will follow a t-distribution with degrees of freedom: \[df = \frac{(s^2_{w}/n_{w} + s^2_{m}/n_{m})^2}{(s^2_{w}/n_{w})^2/(n_{w}-1) + (s^2_{m}/n_{m})^2/(n_{m}-1)}\]

03

Calculate the test statistic and degrees of freedom

\(\bar{x}_{w} = 26.7\), \(s_{w} = 14.1\), \(n_{w} = 40\) \(\bar{x}_{m} = 43.9\), \(s_{m} = 19.4\), \(n_{m} = 40\) Calculate the t-statistic: \[t = \frac{(26.7 - 43.9)}{\sqrt{\frac{14.1^2}{40} + \frac{19.4^2}{40}}} \approx -4.91\] Calculate the degrees of freedom: \[df = \frac{(14.1^2/40 + 19.4^2/40)^2}{(14.1^2/40)^2/(40-1) + (19.4^2/40)^2/(40-1)} \approx 64.24\]

04

Determine the critical value and compare with the test statistic

For a two-tailed test at significance level of \(\alpha=0.05\), the critical values are given by the t-distribution with 64 degrees of freedom: \(t_{\alpha/2}=-1.998\) and \(t_{1-\alpha/2}=1.998\). We can compare the test statistic with the critical values: \(-1.998 > -4.91 < 1.998\)

05

Draw conclusions

Since the t-statistic is less than the lower critical value, we reject the null hypothesis. Therefore, there is convincing evidence that the mean reading level for health-related pages differs for Wikipedia and WebMD at a significance level of \(\alpha=0.05\).

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Key Concepts

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

Understanding Flesch Reading Ease Scores

When exploring the accessibility of written content, especially on the internet, tools like the Flesch reading ease score are invaluable. The score evaluates how easy a text is to understand based on the length of sentences and the number of syllables per word. A higher score indicates a text that's more complicated and challenging to read. In contrast, a lower score suggests the material is simpler and easier to digest.

To calculate the Flesch score, we use the formula:
\[ 206.835 - 1.015 \left(\frac{\text{total words}}{\text{total sentences}}\right) - 84.6 \left(\frac{\text{total syllables}}{\text{total words}}\right) \]
For instance, in the context of the exercise, a lower mean score on Wikipedia suggests that its health-related pages might be more accessible to a broader audience, while a higher mean score on WebMD implies more complex language. Understanding these scores is crucial for content creators and educators who aim to produce material that reaches their intended audience effectively.

The Two-Sample T-Test Explained

When comparing the means of two independent groups, like Wikipedia and WebMD reading levels, statisticians often employ the two-sample t-test. This test helps to determine if the observed differences in sample means are statistically significant or if they could be due to random chance.

The formula for the two-sample t-test is:
\[t = \frac{(\bar{x}_{1} - \bar{x}_{2}) - \Delta}{\sqrt{\frac{s^2_{1}}{n_{1}} + \frac{s^2_{2}}{n_{2}}}}\]
Where:\

\
• \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the sample means,
\
• \(s^2_{1}\) and \(s^2_{2}\) are the sample variances,
\
• \(n_{1}\) and \(n_{2}\) are the sample sizes, and
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• \(\Delta\) represents the hypothesized difference in population means, which is often 0 for testing equality.
\

\
Using this test, we can confidently infer if both websites offer a reading level that's significantly different or if the difference is negligible, thus informing strategies for content creation and design.

Interpreting the Significance Level in Hypothesis Testing

The significance level, commonly denoted as \(\alpha\), is a threshold used to determine the presence of a statistically significant effect. In hypothesis testing, it represents the probability of rejecting the null hypothesis when it is actually true—an error known as a Type I error.

Typically set at 0.05, or 5%, the significance level is a balance between being too lenient (and potentially accepting false positives) and too stringent (risking the rejection of true effects). If the test statistic falls within the critical region defined by this alpha level, the null hypothesis is rejected, indicating that there is a significant difference between the groups in question. In the context of the exercise, by rejecting the null hypothesis at the \(\alpha=0.05\) level, we're concluding with 95% confidence that Wikipedia and WebMD pages have different mean readability scores, suggesting a significant difference in their readability.