Question

The report "How Teens Use Media" (Nielsen, June 2009) says that \(37 \%\) of U.S. teens access the Internet from a mobile phone. Suppose you plan to select a random sample of students at the local high school and will ask each student in the sample if he or she accesses the Internet from a mobile phone. You want to determine if there is evidence that the proportion of students at the high school who access the Internet using a mobile phone differs from the national figure of 0.37 given in the Nielsen report. What hypotheses should you test?

Short answer

The null hypothesis (H0) should be that the proportion of high school students who access the internet using a mobile phone is equal to the national proportion (0.37). The alternative hypothesis (Ha) should be that the proportion of high school students who access the internet using a mobile phone is not equal to the national portion (0.37).

Step-by-step solution

Identify Null and Alternative Hypotheses

The null hypothesis (H0) is an assumption that the population proportion is equal to a specified value, in this case, the national figure of 0.37. The alternative hypothesis (Ha) is the opposite of the null hypothesis, meaning that the population proportion is not equal to 0.37. Therefore, H0: p = 0.37 and Ha: p ≠ 0.37.

Key concepts

Null Hypothesis

Understanding the null hypothesis is crucial in hypothesis testing. It is a statement that assumes no effect or no difference in a research study. Specifically, it is the presumption that the population proportion you are examining is the same as the reference group or a specific value, which is often derived from previous research or theoretical expectations.

For example, if we are considering whether a certain percentage of high school students access the internet from their mobile phones, the null hypothesis would assert that the proportion in the high school is the same as the national figure, which could be, let's say, 37%. In formal terms, the null hypothesis is expressed as \(H_0: p = 0.37\), where \(p\) represents the population proportion. When you perform a hypothesis test, your goal is to determine whether the evidence collected (i.e., the data from your sample) is strong enough to reject the null hypothesis.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. It is what you aim to support with your data collection and analysis. In the context of our example regarding high school students accessing the internet via mobile phones, the alternative hypothesis suggests a difference from the national figure. The alternative hypothesis, denoted as \(H_a\) or \(H_1\), posits that the true population proportion is not 37%.

It is expressed mathematically as \(H_a: p eq 0.37\). If evidence supports the alternative hypothesis, you may conclude a difference exists. However, it's important to note that 'supporting' the alternative hypothesis does not confirm it as true; it implies that the data you have collected is not consistent with the null hypothesis, prompting the need for further research or inquiry.

Population Proportion

Population proportion refers to the percentage of individuals in a population who exhibit a particular characteristic. In the example given, it would represent the proportion of all students at the high school who access the internet from a mobile phone. The population proportion is often denoted as \(p\) and is a critical parameter in statistical hypothesis testing.

In order to infer something about the population proportion, we collect data from a sample. This sample should be representative of the population to ensure that we get an unbiased estimate of \(p\). Once we have the sample data, we can calculate the sample proportion and use it to perform hypothesis tests, comparing our findings against the established or claimed population proportion.

Statistical Significance

Statistical significance is a determination of whether the observed differences in data are due to chance or represent true differences in the population. It answers the question: 'Based on the sample, can we infer that the true population proportion differs significantly from the claimed value?'

When testing our hypotheses, we calculate a p-value, which reflects the probability of obtaining our observed results if the null hypothesis were true. A small p-value (usually less than 0.05) indicates that finding such a sample is highly unlikely if the null hypothesis is true, leading to the conclusion that the observed difference is statistically significant. At this point, the null hypothesis is typically rejected, favoring the alternative hypothesis that suggests a real difference in population proportions exists.