Performing a Sign Test In Exercises 7–22, (c) find the test statistic,
Social Media A research group claims that the median age of the users of a social media website is greater than 30 years old. In a random sample of 24 users, 11 are less than 30 years old, 10 are more than 30 years old, and 3 are 30 years old. At , can you support the research group’s claim? (Adapted from Pew Research Center)

Verified step by step guidance

1. Step 1: Understand the problem. The claim is that the median age of users of a social media website is greater than 30 years old. This is a one-tailed test since the claim specifies a direction ('greater than'). The data includes 24 users: 11 are less than 30 years old, 10 are more than 30 years old, and 3 are exactly 30 years old.
2. Step 2: Exclude the data points where the age is exactly 30 years old, as these do not contribute to the sign test. This leaves 21 users: 11 less than 30 years old and 10 more than 30 years old. Assign a '+' sign to values greater than 30 and a '-' sign to values less than 30.
3. Step 3: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The median age of users is 30 years old. H₁: The median age of users is greater than 30 years old. This is a one-tailed test.
4. Step 4: Calculate the test statistic. The test statistic for the sign test is the smaller of the number of '+' signs or '-' signs. In this case, there are 10 '+' signs and 11 '-' signs. The test statistic is the smaller of these two values.
5. Step 5: Determine the critical value and compare it to the test statistic. Use the binomial distribution with n = 21 and p = 0.5 (under the null hypothesis) to find the critical value for the given significance level (α). If the test statistic is less than or equal to the critical value, fail to reject H₀. Otherwise, reject H₀ and support the claim that the median age is greater than 30 years old.

Key Concepts

Sign Test

The Sign Test is a non-parametric statistical method used to determine if there is a significant difference between the median of a sample and a hypothesized value. It is particularly useful when the data does not meet the assumptions required for parametric tests, such as normality. In this test, the signs of the differences between each observation and the hypothesized median are counted, allowing researchers to assess whether the median is significantly different from the hypothesized value.

Hypothesis Testing

Hypothesis testing is a statistical procedure that allows researchers to make inferences about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject the null hypothesis. In this context, the null hypothesis would state that the median age of users is 30 years or less, while the alternative hypothesis would assert that it is greater than 30 years.

Median

The median is a measure of central tendency that represents the middle value of a dataset when it is ordered from least to greatest. It is particularly useful in skewed distributions, as it is less affected by outliers than the mean. In the context of the question, understanding the median age of social media users is crucial for evaluating the research group's claim about the user demographics.