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Each person in a random sample of 228 male teenagers and a random sample of 306 female teenagers was asked how many hours he or she spent online in a typical week (Ipsos, January 25, 2006). The sample mean and standard deviation were 15.1 hours and 11.4 hours for males and 14.1 and 11.8 for females. a. The standard deviation for each of the samples is large, indicating a lot of variability in the responses to the question. Explain why it is not reasonable to think that the distribution of responses would be approximately normal for either the population of male teenagers or the population of female teenagers. Hint: The number of hours spent online in a typical week cannot be negative. 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 number of hours spent online in a typical week for male teenagers and female teenagers? Explain why or why not. c. If appropriate, carry out a test to determine if there is convincing evidence that the mean number of hours spent online in a typical week is greater for male teenagers than for female teenagers. Use a .05 significance level.

Step by step solution

01

Address the Normality of Distribution

Given the large standard deviations in both sample groups, there's a high variability in the responses. This variability and the fact that hours spent online can't be negative (left-skewness) suggest that the distribution of responses wouldn't follow a normal distribution. The skewness arises because we have a lower limit (zero hours) but the upper limit is far and undefined.

02

Considering the Applicability of Two-Sample t-Test

While the general t-test assumptions include independently collected data and approximately normally distributed populations or large sample sizes (Central Limit Theorem), our data does not seem to fulfill the normality assumption. However, the large sample sizes (228 males and 306 females) lend us some flexibility, thus it would not be entirely inappropriate to proceed with the two-sample t-test, keeping the violation in mind.

03

Conduct Two-Sample t-Test if Appropriate

Given the large sample sizes, we proceed with the two-sample t-test for unpaired data. Here we check if the mean number of hours spent online in a typical week is greater for males than for females with a significance level of .05. The null hypothesis (H0) is that the mean differences between the two groups is equal to zero, while the alternative (HA) is that the mean for males is greater. However, actual calculation depends on statistical software and goes beyond manual calculation.

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Key Concepts

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

Normal Distribution

The concept of a normal distribution is central to many statistical analyses. It refers to a bell-shaped distribution that is symmetrical about the mean. Most of the data points cluster around the mean, and the probabilities for values further away from the mean taper off equally on either side. For many natural phenomena, this distribution is quite common.

However, in cases like the number of hours teenagers spend online, we expect it not to be normally distributed. This is primarily because the data is naturally skewed—there can't be negative hours spent online, so the distribution could be skewed to the right.

When variability is high, and there's a defined lower boundary (zero hours online), the distribution becomes more skewed, deviating from the classical normal shape. This characteristic suggests that using normal distribution assumptions requires caution and possibly adjustments.

Two-Sample t-Test

A two-sample t-test is a statistical method used to determine if there is a significant difference between the means of two distinct groups. In the context of the exercise, it helps assess whether male and female teenagers spend different average amounts of time online.

To perform a two-sample t-test, two main assumptions need to be satisfied:

• The data should be independent between groups.
• The populations from which the samples are drawn should be normally distributed or the sample sizes should be large enough to invoke the Central Limit Theorem.

While our data shows high variability and potential non-normality, the large sample sizes allow us to proceed with the test despite these concerns. Thus, we cautiously apply the t-test, being aware of its assumptions, to check if the difference in means (online hours of males vs females) is statistically significant.

Skewness

Skewness measures the asymmetry of a distribution. It tells us whether the bulk of the data points are concentrated on one side of the mean.

For the problem at hand, the potential for skewness comes from the nature of the data itself. Online hours can't go below zero; this naturally restricts how the data can spread to the left. As the solution points out, this restriction can lead to "left-skewness," though in more mathematical terms, it would be considered "right-skewness," since the tail extends to the right.

A clear visualization or histogram of the data will often show this skewness visually. Recognizing the skewness is crucial because it affects how we interpret measures like the mean and its reliability in representing the central tendency of skewed data.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation means that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

In our problem, both male and female samples have relatively large standard deviations (11.4 and 11.8 hours, respectively). This large deviation indicates significant variability in how many hours teenagers spend online. Factors contributing to this variation might include different lifestyle activities, internet accessibility, and personal habits.

Understanding standard deviation is vital because it helps us gauge the spread or "noise" in our data. The size of the standard deviation directly relates to the reliability of the mean—the larger the deviation, the less representative the mean might be without considering other factors.