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Internet addiction has been described as excessive and uncontrolled Internet use. The authors of the paper "Gender Difference in the Relationship Between Internet Addiction and Depression" (Computers in Human Behavior [2016]: \(463-470\) ) used a score designed to measure the extent and severity of Internet addiction in a study of 836 male and 879 female sixth grade students in China. Internet Addiction was measured using Young's Internet Addiction Diagnostic Test. The lowest possible score on this test is zero, and higher scores indicate higher levels of Internet addiction. For the sample of males, the mean Internet Addiction score was 1.51 and the standard deviation was \(2.03 .\) For the sample of females, the mean was 1.07 and the standard deviation was \(1.63 .\) For purposes of this exercise, you can assume that it is reasonable to regard these two samples as representative of the population of male Chinese sixth grade students and the population of female Chinese sixth grade students, respectively. a. The standard deviation is greater than the mean for each of these samples. Explain why it is not reasonable to think that the distribution of Internet Addiction scores would be approximately normal for either the population of male Chinese sixth grade students or the population of female Chinese sixth grade students. b. Given your response to Part (a), would it be appropriate to use the two- sample \(t\) test to test the null hypothesis that there is no difference in the mean Internet Addiction score for male Chinese sixth grade students and female Chinese sixth grade students? Explain why or why not. c. If appropriate, carry out a test to determine if there is convincing evidence that the mean Internet Addiction score is greater for male Chinese sixth grade students than for female Chincse sixth grade students. Use \(\alpha=0.05\).

Step by step solution

01

Evaluate normality of score distributions

The distribution of Internet Addiction scores is said to be approximately normal if the standard deviation is not greater than the mean (since the lowest possible score is zero, and higher scores indicate higher levels of Internet addiction). For the male sample, the mean is 1.51, and the standard deviation is 2.03. For the female sample, the mean is 1.07, and the standard deviation is 1.63. In both samples, the standard deviation is greater than the mean, which means it is not reasonable to assume that the distribution of scores is approximately normal for either population.

02

Assess appropriateness of using a two-sample t-test

Since the assumption of normality is not satisfied for both populations, it would not be appropriate to use a two-sample t-test to test the null hypothesis that there is no difference in the mean Internet Addiction score between male and female Chinese sixth grade students.

03

Carry out a test, if appropriate

As discussed in Step 2, it is not appropriate to carry out a t-test due to the violation of normality assumption. Therefore, we cannot proceed with any further testing. In conclusion, based on the given data, it is not appropriate to use a two-sample t-test to determine if there is a difference in mean Internet Addiction scores between male and female Chinese sixth grade students since the assumption of normality is not met for the score distributions.

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Key Concepts

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

Internet Addiction Diagnostic Test

The Internet Addiction Diagnostic Test is a vital tool for examining the extent of an individual's addiction to the internet. Developed by Dr. Kimberly Young, this test allows for a structured assessment by providing a series of questions that gauge the impact of internet usage on daily life. Scoring is done based on the respondents' agreement with statements that evaluate factors such as preoccupation with the internet, loss of control over usage, and the presence of withdrawal symptoms when not online.

As exhibited in the study with Chinese sixth grade students, higher scores signify a stronger presence of addictive behaviors associated with internet use. This score is an aggregate measure but can vary significantly among different populations. Therefore, understanding the context and demographic specifics is crucial when interpreting results. In cases where scores deviate significantly from the expected mean or show extreme variability, it suggests a need for deeper investigation before drawing broad conclusions about internet addiction levels within that population.

Statistical Normality

Statistical normality is a key concept in data analysis that underpins many statistical tests, including the t-test. A normal distribution, often known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In real-world scenarios, many continuous variables are commonly normally distributed.

However, when the standard deviation is larger than the mean, as seen in the study with Chinese sixth graders, it suggests substantial skewness or outliers in the data. This deviation from normality can significantly impact the validity of statistical tests like the t-test that assume normal distribution of the data. For educational purposes, it's crucial to recognize that when normality is questioned, alternative non-parametric tests or data transformation techniques might be suitable to accurately analyze the data.

Two-Sample T-Test

The two-sample t-test is a statistical method used to determine if there are significant differences between the means of two independent samples. For instance, this would be suitable to compare the mean internet addiction scores between male and female students if the data were normally distributed.

However, the test requires two primary assumptions to be met for validity: the distributions of the two groups should be approximately normal, and the variances of the groups should be equal (homoscedasticity). When the assumption of normality is violated, as seen with the Internet Addiction scores of the Chinese students, the two-sample t-test may yield unreliable results, and statistical inferences drawn from it could be misleading. Alternatives like the Mann-Whitney U test, which does not require the data to be normally distributed, might be used instead in such scenarios. It is essential to assess the suitability of statistical tests before application to ensure the integrity of research findings.