Problem 15 - An Introduction to Mathematical Statistics and Its Applications

Question

To measure the effect on coordination associated with mild intoxication, thirteen subjects were each given \(15.7 \mathrm{~mL}\) of ethyl alcohol per square meter of body surface area and asked to write a certain phrase as many times as they could in the space of one minute (127). The number of correctly written letters was then counted and scaled, with a scale value of 0 representing the score a subject not under the influence of alcohol would be expected to achieve. Negative scores indicate decreased writing speeds; positive scores, increased writing speeds. Use the signed rank test to determine whether the level of alcohol provided in this study had any effect on writing speed. Let \(\alpha=0.05 .\) Omit Subject 8 from your calculations. \begin{tabular}{crrr} \hline Subject & Score & Subject & Score \\ \hline 1 & \(-6\) & 8 & 0 \\ 2 & 10 & 9 & \(-7\) \\ 3 & 9 & 10 & 5 \\ 4 & \(-8\) & 11 & \(-9\) \\ 5 & \(-6\) & 12 & \(-10\) \\ 6 & \(-2\) & 13 & \(-2\) \\ 7 & 20 & & \\ \hline \end{tabular}

Step-by-Step Solution

Answer

The conclusion on whether the level of alcohol provided in the study had any effect on writing speed is drawn by comparing the test statistic and the critical value. If the test statistic is less than or equal to the critical value, there is evidence to suggest that alcohol did affect the writing speed at a 0.05 significance level. The specific values can only be determined by going through the steps involving computation of differences, ranking, and look-up of the critical value from the Signed Rank Test table.

Step 1: Compute Differences

Since we are checking the effect of alcohol, we need to calculate the difference between the scores and the expected score of a sober person, which is 0. However, Subject 8 is to be omitted.

Step 2: Compute Absolute Differences and Assign Ranks

Find the absolute differences of the scores and rank them in increasing order. Scores with the same absolute difference should be assigned the average of the ranks they would have occupied if they were different.

Step 3: Compute Sum of Ranks

Add up the ranks corresponding to negative and positive differences separately. The smaller sum is the test statistic, denoted as T.

Step 4: Determine Critical Value

Use a statistical table for the Signed Rank Test to find the critical value at \(\alpha = 0.05\) for 12 observations (since we omitted one subject).

Step 5: Compare Test Statistic with Critical Value

If T is less than or equal to the critical value, reject the null hypothesis (i.e., there is no effect of alcohol on writing speed). Otherwise, do not reject the null hypothesis.

Key Concepts

Signed Rank Test

The Signed Rank Test is a non-parametric statistical method used to evaluate the median of a distribution. It’s often applied when you want to compare the differences in matched pairs or related samples. Unlike parametric tests, it doesn't assume a normal distribution, making it a flexible choice for skewed data or small sample sizes.

Here's how it works:

• Calculate the difference between each pair of observations.
• Ignore zero differences, focusing on non-zero ones.
• Rank the absolute values of these differences, but preserve their signs.
• The Signed Rank Test statistic is the sum of the ranks corresponding to the less frequent sign, whether positive or negative.

This test helps in determining whether a particular treatment or condition, like alcohol consumption in this case, has an effect.

Hypothesis Testing

Hypothesis Testing is a statistical method used to assess the validity of a claim about a population parameter based on sample data. The process begins by stating two opposing hypotheses: the null and the alternative.

We then calculate a test statistic and compare it to a critical value from a relevant statistical distribution. If our statistic falls beyond this critical region, we have sufficient evidence to reject the null hypothesis. Otherwise, we do not reject it, acknowledging there's not enough evidence to support the alternative hypothesis. Hypothesis testing provides a structured approach to making decisions or inferences in research.

Null Hypothesis

The Null Hypothesis, often denoted as \(H_0\), is a statement that there is no effect or no difference in a certain context.

In the context of the Signed Rank Test, the null hypothesis asserts that alcohol has no effect on writing speed.

• We assume the null hypothesis as true initially.
• The goal is to determine whether the observed data provides enough evidence to reject it in favor of an alternative hypothesis.
• In the given exercise, failing to reject the null hypothesis would mean that any observed changes in writing speed might still coincide with natural variation rather than being a result of alcohol consumption.

The concept acts as a foundation upon which hypothesis testing is established, guiding the statistical inquiry.

Significance Level

The Significance Level, represented by \(\alpha\), is the threshold for deciding whether an observed effect is statistically significant. It sets the probability of making a Type I error, which is rejecting a true null hypothesis.

In most studies, including the current exercise, a significance level of 0.05 is often used. This means there's a 5% risk of concluding that there's an effect when there actually isn't one.

• If the p-value of the test statistic is less than or equal to \(\alpha\), we reject the null hypothesis.
• If it's greater, we do not reject it.

Choosing an appropriate significance level depends on the context of the study, as more critical decisions might warrant a stricter threshold, such as 0.01.