Question

An article titled "Teen Boys Forget Whatever It Was" appeared in the Australian newspaper The Mercury (April 21, 1997). It described a study of academic performance and attention span and reported that the mean time to distraction for teenage boys working on an independent task was 4 minutes. Although the sample size was not given in the article, suppose that this mean was based on a random sample of 50 teenage boys and that the sample standard deviation was 1.4 minutes. Is there convincing evidence that the average attention span for teenage boys is less than 5 minutes? Test the relevant hypotheses using \(\alpha=0.01\).

Short answer

Based on the t-test, the average attention span for teenage boys is less than 5 minutes at the \( \alpha=0.01 \) level of significance.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (H0) is: \( \mu = 5 \), which means the average attention span for teenage boys is assumed to be 5 minutes. The alternative hypothesis (Ha) is: \( \mu < 5 \), suggesting the average attention span for these boys is less than 5 minutes.

Calculate the test statistic (t-score)

The t-score is calculated by using the formula: \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \), where \( \bar{x} \) is the sample mean (4 mins), \( \mu \) is the population mean under null hypothesis (5 mins), s is the standard deviation (1.4 mins) and n is the sample size (50). Substituting: \( t = \frac{4 - 5}{\frac{1.4}{\sqrt{50}}} = -5.35 \).

Determine the critical value

The critical value for a left-tailed test with a significance level of \( \alpha=0.01 \) and degrees of freedom (df = n-1 = 49) can be found from a standard t-distribution table or using a t-value calculator. For this problem, the critical value is approximately -2.40.

Compare t-score with critical value

If the calculated t-value is less than the critical value, then reject the null hypothesis. In this case, -5.35 < -2.40. Therefore, we reject the null hypothesis.

Conclusion

Since the null hypothesis was rejected, there is convincing evidence that the average attention span for teenage boys is less than 5 minutes.

Key concepts

Null Hypothesis

The null hypothesis is the starting point in hypothesis testing. It assumes that there is no effect or no difference and that any observed variation is due to chance. In our exercise, the null hypothesis (\( H_0 \)) is presented as \( \( \mu = 5 \) \), meaning we initially assume the average attention span for teenage boys is 5 minutes.

When we set out to conduct a statistical test, we are effectively trying to find enough evidence to refute the null hypothesis. It's like saying you think everything is normal until proven otherwise. If sufficient evidence against the null hypothesis is found (often through calculating p-values or comparing test statistics to a critical value), we reject it and accept that our sample suggests something different might be happening.

Alternative Hypothesis

The alternative hypothesis (\( H_a \) or \( H_1 \)) represents what the researcher is trying to support; it is a statement that contradicts the null hypothesis. For the exercise, the alternative hypothesis is \( \( \mu < 5 \) \), proposing that the average attention span is actually less than 5 minutes.

Unlike the null hypothesis, the alternative hypothesis is what you suspect might be the truth behind the data. It’s a critical component of hypothesis testing, offering a new theory proposed as the true effect or relationship if evidence suggests the null hypothesis is unlikely. Demonstrating the alternative can be true has significant implications, especially when establishing new theories or challenging existing beliefs.

T-Score Calculation

The t-score, or t-statistic, is a ratio that compares the difference between the sample mean and the population mean to the amount of variability in the sample. It's calculated using the formula \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \), where \( \bar{x} \) is the sample mean, \( \( \mu \) \) represents the population mean assumed under the null hypothesis, \( s \) stands for the sample standard deviation, and \( n \) is the sample size.

The calculated t-score for our exercise was -5.35. This value tells us how many standard deviations the sample mean is away from the population mean. Since this t-score is negative, it points to the sample mean being less than the population mean—which in the context of our hypothesis test suggests that the attention span may indeed be shorter than expected.

Critical Value

The critical value is a threshold to which the test statistic is compared to determine whether to reject the null hypothesis. It's derived from the probability distribution of the test statistic under the assumption that the null hypothesis is true and is based on the desired level of significance, denoted by \( \alpha \).

In our exercise, the given significance level is \( \alpha=0.01 \), indicating a 1% chance of wrongly rejecting the null hypothesis when it's actually true (a type I error). Using this significance level, we find a critical value from a t-distribution table which, for 49 degrees of freedom, is approximately -2.40. Our calculated t-score of -5.35 is less than this critical value, indicating a statistically significant result, and we reject the null hypothesis.

Sample Standard Deviation

Sample standard deviation (\( s \) ) measures the dispersion or variability within a sample of data. It tells us how much the individual data points differ from the sample mean. A high standard deviation means the values are spread out over a wider range, while a low standard deviation indicates they're clustered closely around the mean.

In our problem, the sample standard deviation is 1.4 minutes. This value is crucial in calculating the t-score as it is used to standardize the difference between the sample mean and the population mean. The calculation accounts for the size of the sample, expressing the variability in terms of the average distance from the mean for a single observation. Understanding standard deviation is fundamental, as it impacts the t-score and, ultimately, the conclusions drawn from hypothesis tests.