91Ó°ÊÓ

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

Reasoning about the Data Distribution

Considering that the number of hours spent online cannot be negative, it is not reasonable to assume a normal distribution. The high standard deviation indicates a great variability in responses, and with the lower limit of zero hours (since negative hours are impossible), the distribution is likely skewed towards the right.

02

Assessment of the T-test Usage

The two-sample t-test assumes that the populations from which the samples are drawn are normally distributed. Given the assumption made in Step 1, it is not entirely appropriate to use a two-sample t-test. However, when the sample sizes are sufficiently large (as in this case), the Central Limit Theorem allows us to use a t-test.

03

Hypothesis Testing

You should carry out a hypothesis testing given that the law of large numbers applies due to the large sample sizes. First, define null and alternative hypotheses. The null hypothesis (\(H_0\)): the mean time online for males equals the mean time online for females. The alternative hypothesis (\(H_A\)): the mean time online for males differs from females. Then, calculate the test statistic using the formula: \[t = \frac{(\bar{x_m} - \bar{x_f}) - 0}{\sqrt{(s_m^2/n_m) + (s_f^2/n_f)}}\]. Substituting with the given values, the calculated t value can be compared to the critical t value for one-tailed t-test with α = 0.05. If the calculated t is greater, we reject the null hypothesis. Else we cannot reject.

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Key Concepts

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

Two-Sample T-Test

A two-sample t-test is a statistical procedure used to determine whether there is a significant difference between the means of two independent groups. In our case, the goal is to see if male teenagers spend more time online compared to female teenagers. The test compares the sample means of these two groups and evaluates if the difference observed is due to random chance or if it is statistically significant.

Before conducting a two-sample t-test, there are some important assumptions:

• The populations from which each sample is taken should be normally distributed.
• The samples should be independent of each other.
• The variance within each population should be approximately equal.

Even though our distribution may not be perfectly normal due to the constraint of non-negative hours, the large sample sizes justify using a t-test under the Central Limit Theorem, which smooths out deviations from normality.

Central Limit Theorem

The Central Limit Theorem (CLT) is a key principle in statistics. It states that the sampling distribution of the sample mean will tend to be approximately normal, regardless of the shape of the population distribution, provided the sample size is sufficiently large. In simpler terms, even if the underlying population distributions are not normal, the means from large samples will tend to become normally distributed.

This theorem is essential when dealing with real-world data, which often does not follow a perfect normal distribution. In our scenario with teenage online hour data, even though individual responses are not normally distributed (they are skewed because hours cannot be negative), the Group means, due to the large sample sizes of 228 males and 306 females, follow the normal distribution closely enough to use a two-sample t-test confidently.

Normal Distribution

A normal distribution is a bell-shaped frequency distribution graph which is symmetric around its mean. Normal distributions are important in statistics because they have properties that make them sensible approximations of various data sets and convenient for analytical methods.

However, in our exercise, the hours spent online cannot be negative, creating a natural skew in the data. With large standard deviations for both male and female teenagers, the distribution of hours spent online likely deviates from a perfect normal distribution, possibly skewed to the right. Despite this, because the sample sizes are large, the Central Limit Theorem allows us to assume a normal distribution for the purposes of our hypothesis test.

Standard Deviation

Standard deviation is a measure of the dispersion, or spread, of a set of values. A higher standard deviation indicates a greater range of values in the dataset, while a lower standard deviation indicates that the values tend to be close to the mean.

In our exercise, the standard deviation for male teenagers is 11.4 hours and for female teenagers, it is 11.8 hours. These numbers show considerable variability in how much individual teenagers use the internet weekly. Such variability can influence data interpretation and test results because it reflects the reliability of the mean as a representative measure. Nonetheless, the size of our samples helps offset some concerns about this variability, as larger sample sizes tend to provide more reliable estimated means.