Large Sample and a Small Difference It has been said that with really large samples, even very small differences between the sample mean and the claimed population mean can appear to be significant, but in reality they are not significant. Test this statement using the claim that the mean IQ score of adults is 100, given the following sample data: n = 1,000,000, x_bar = 100.05, s = 15 . Based on this sample, is the difference between x_bar = 100.05 and the claimed mean of 100 statistically significant? Does that difference have practical significance?

Verified step by step guidance

1. Step 1: Formulate the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 100 (the population mean is 100), and the alternative hypothesis is H₁: μ ≠ 100 (the population mean is not 100).
2. Step 2: Calculate the test statistic using the formula for the z-test for a single sample mean: z = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. Substitute the given values: x̄ = 100.05, μ = 100, s = 15, and n = 1,000,000.
3. Step 3: Determine the critical z-value(s) for the chosen significance level (commonly α = 0.05 for a two-tailed test). Use a z-table or statistical software to find the critical values corresponding to α/2 in each tail.
4. Step 4: Compare the calculated z-value from Step 2 to the critical z-value(s) from Step 3. If the calculated z-value falls outside the range defined by the critical z-values, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
5. Step 5: Assess practical significance. Even if the null hypothesis is rejected (indicating statistical significance), consider whether the observed difference (x̄ - μ = 0.05) is meaningful in a real-world context. Given the large sample size, very small differences can appear statistically significant but may lack practical importance.

Key Concepts

Statistical Significance

Statistical significance refers to the likelihood that a relationship between variables in a sample is not due to random chance. It is typically assessed using a p-value, which indicates the probability of observing the data if the null hypothesis is true. A common threshold for significance is p < 0.05, meaning there is less than a 5% chance that the observed difference is due to random variation.

Practical Significance

Practical significance assesses whether a statistically significant result has real-world relevance or meaningful impact. While a result may be statistically significant, it does not necessarily imply that the difference is large enough to be of practical importance. For example, a small difference in means may be statistically significant in a large sample but may not have any meaningful implications in practice.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution. This theorem is crucial when working with large samples, as it allows for the use of normal distribution properties to make inferences about population parameters, even when the underlying data may not be normally distributed.