Question

When \(n>25\), what is used in place of Table J for the sign test?

Short answer

For \( n > 25 \), the normal approximation to the binomial distribution is used instead of a table.

Step-by-step solution

Understanding the Problem

We are asked to determine what is used for the sign test when the sample size \( n > 25 \). The sign test is a non-parametric test used to determine if there is a significant difference between the medians of two related groups.

Familiarize with Sign Test

The sign test for small samples typically uses a specific table (like Table J) to determine the critical value needed to decide if we reject the null hypothesis. This table is composed based on sample size and significance level.

Identify Solution for Large Samples

For larger samples, specifically when \( n > 25 \), tables like Table J are impractical due to the extensive number of rows. Instead, the normal approximation is used. This method employs the properties of the normal distribution to approximate the critical value instead of directly using a table.

Apply Normal Approximation

The normal approximation is applied by using the Z-score derived from the normal distribution. This involves calculating the mean and standard error based on the given signs in the sample and comparing it against the standard normal distribution to decide rejection of the null hypothesis.

Key concepts

Understanding Non-Parametric Test

Non-parametric tests are statistical methods often used when we do not want to make assumptions about the population’s distribution from which our sample is drawn. Unlike parametric tests, they do not rely on data belonging to any specific distribution, such as a normal distribution. This makes them particularly useful in cases where data doesn’t meet the assumptions needed for parametric procedures. Non-parametric tests also handle outliers and skewed data more flexibly. The sign test is a perfect example of a non-parametric test. It looks at the signs of the differences between paired observations, rather than their numeric magnitude. The advantage? You can use it to assess the median difference in two related samples without assuming a normal distribution. Essentially, non-parametric tests grant more freedom when analyzing real-life data that might not fit into neat, predictable patterns.

Using Normal Approximation in Sign Test

When working with larger samples, such as when the sample size exceeds 25, using a table for critical values can become cumbersome. This is where normal approximation steps in. It leverages the Central Limit Theorem, which states that the distribution of the sample mean approaches a normal distribution as the sample size grows, even if the original data wasn't normally distributed. In the context of the sign test, normal approximation allows us to replace a tedious table look-up with a simpler calculation.

• Determine the number of positive and negative differences in your sample.
• Calculate the expected number of positive signs under the null hypothesis.
• Compute the standard error.
• Use these to find a Z-score, which will help determine how unusual your data is under the assumption of no median difference.

By utilizing the Z-score, you can compare the result against a standard normal distribution, simplifying decision-making about the null hypothesis without sifting through lengthy statistical tables.

Critical Value Determination in Sign Test

In hypothesis testing, determining the critical value is crucial as it helps you decide whether to reject the null hypothesis. For small samples, tables (like Table J) provide these values directly based on sample size and significance level. These tables essentially tell you how extreme your test statistic needs to be for you to consider it unlikely under the null hypothesis. However, with larger samples, calculating this directly becomes unwieldy. Thus, critical value determination shifts to using normal approximation. You compute a Z-score as a way of finding this critical threshold, comparing it against standard normal distribution tables to decide if your test statistic falls in the rejection region. Through this method, even with larger datasets, you're able to effectively use the sign test’s principles and make informed statistical decisions.