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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

Understanding the Standard Deviations and Normal Distribution

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 because the standard deviation is quite large compared to the mean. This large standard deviation indicates a considerable variation in the data. Moreover, since the number of hours spent online cannot be negative, the distribution could likely be skewed due to no negative values being possible.

02

Appropriateness of Two-Sample t-Test

The two-sample t-test is not the best method to use for this case. This test assumes that the data are approximately normally distributed for valid interpretation of results. Here, we've established in the previous step that it is not reasonable to assume a normal distribution. Thus, a t-test may not provide the most accurate results. Another method to test the null hypothesis, such as a non-parametric test (like the Mann-Whitney U test), may be more appropriate.

03

Performing a test (if applicable)

Since in step 2 we have assessed that the t-test is not appropriate due to the data not being normally distributed, we do not perform a t-test in this case. To ascertain if there is a significant difference in the mean number of hours spent online by male and female teenagers, a non-parametric test should be used.

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Key Concepts

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

Random Sampling

Understanding the concept of random sampling is crucial when collecting data to ensure its representative nature. In a random sample, every individual in the population has an equal chance of being selected. This process plays a pivotal role in reducing biases, thus enhancing the accuracy of inferences made regarding broader populations. For example, in the educational study mentioned, samples of male and female teenagers were randomly selected to avoid favoritism and other biases that could skew the results.

Standard Deviation

The term standard deviation refers to a statistical measure that reflects the amount of variation or dispersion in a set of values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation signifies that data points are spread out over a wider range of values.

In our educational scenario involving teenagers' online hours, the sizeable standard deviations found suggest considerable variability among individuals' online habits, indicating inconsistent behaviors within the groups.

Normal Distribution

Normal distribution, often illustrated as a bell-shaped curve, describes a situation where data tends to be symmetrically distributed around the mean. In an ideal normal distribution, most data points cluster around the central peak, and the probabilities for values taper off symmetrically on both sides.

However, in instances such as the teenager study, large standard deviations and restrictions on data (like the impossibility of having negative hours online) challenge the assumption of normality. This highlights that not all datasets will follow a normal distribution, particularly when there are natural boundaries to data.

Two-Sample t-Test

The two-sample t-test is a statistical method used to determine whether two sample means are significantly different from each other. It assumes that the data from both samples are approximately normally distributed and have similar variances. When data do not satisfy these assumptions, results from the t-test may not be reliable.

In the study of online hours by teenagers, since the data does not appear to be normally distributed, relying upon the two-sample t-test might lead to incorrect conclusions. This highlights the importance of selecting the appropriate statistical test based on the data's characteristics.

Significance Level

The significance level, often represented by the Greek letter alpha (α), is a threshold used to decide whether to reject the null hypothesis. It is the probability of making the error of rejecting a true null hypothesis (Type I error). A common significance level used is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none.

Tying this to the teenagers' study, if a proper test (like a non-parametric test) were used and results in a p-value less than 0.05, it would suggest evidence of a true difference between the online hours of male and female teenagers, beyond random chance.

Non-Parametric Test

Non-parametric tests are statistical tests that do not assume a specific distribution for the data. They are particularly useful when data do not meet the assumptions required for parametric tests, such as normality. The Mann-Whitney U test, for example, is a non-parametric test that can compare medians from two independent samples.

In our example, employing a non-parametric test would be recommended to analyze the difference in weekly online hours between genders, as it circumvents the need for data to fit a normal distribution, thus providing a more accurate reflection of the situation.