Feeling Your Age A research organization conducts a survey by randomly selecting adults and asking each, “How do you feel relative to your age?” The results are shown in the figure. (Adapted from Pew Research Center)
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a. Use a sign test to test the null hypothesis that the proportion of adults who feel older is equal to the proportion of adults who feel younger. Assign a + sign to each adult who responded “older,” assign a - sign to each adult who responded “younger,” and assign a 0 to each adult who responded “my age.” Use α = 0.05

Verified step by step guidance

1. Step 1: Understand the problem and the hypothesis. The null hypothesis (H₀) states that the proportion of adults who feel older is equal to the proportion of adults who feel younger. The alternative hypothesis (H₁) states that these proportions are not equal. This is a two-tailed test.
2. Step 2: Assign signs to the responses. Assign a '+' sign to each adult who responded 'older,' a '-' sign to each adult who responded 'younger,' and a '0' to each adult who responded 'my age.' Exclude the '0' responses from the analysis since they do not contribute to the comparison.
3. Step 3: Count the number of '+' and '-' signs. Let n₊ represent the number of '+' signs and n₋ represent the number of '-' signs. The total number of non-zero responses is n = n₊ + n₋.
4. Step 4: Use the binomial distribution to calculate the test statistic. Under the null hypothesis, the probability of a '+' or '-' sign is equal (p = 0.5). The test statistic is the smaller of n₊ or n₋. Calculate the probability of observing this test statistic or something more extreme using the binomial probability formula: P(X ≤ k) = Σ (n choose k) * p^k * (1-p)^(n-k), where k is the test statistic.
5. Step 5: Compare the p-value to the significance level α = 0.05. If the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether there is evidence to suggest that the proportions of adults who feel older and younger are different.

Key Concepts

Sign Test

The sign test is a non-parametric statistical method used to determine if there is a significant difference between paired observations. It involves assigning signs (+, -, 0) to responses based on their direction relative to a median or hypothesized value. In this case, it assesses whether the number of adults feeling older differs from those feeling younger, making it suitable for ordinal data or when the sample size is small.

Null Hypothesis

The null hypothesis (H0) is a statement that there is no effect or no difference, serving as a starting point for statistical testing. In this scenario, the null hypothesis posits that the proportion of adults feeling older is equal to those feeling younger. Testing this hypothesis allows researchers to determine if observed differences in survey responses are statistically significant or due to random chance.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. Commonly set at 0.05, it indicates a 5% risk of concluding that a difference exists when there is none. In this analysis, if the p-value from the sign test is less than 0.05, it suggests that the feelings of age among adults are significantly different, warranting further investigation.