On a standard anagram task, people successfully complete an average µ=26 anagrams with a standard deviation=4. This distribution is normal.

A researcher would like to demonstrate that the arousal from anxiety is distracting and will decrease task performance. A sample of n=14 anxiety-ridden participants is tested on the task.

The average number of anagrams solved is M=23.36. Do the anxiety-ridden participants show a decrease in task performance?

To determine if anxiety-ridden participants show a decrease in task performance, we can use a statistical test called a one-sample t-test. This test allows us to compare the mean of a single sample to a known population mean and determine if there is a significant difference.

Here's how we can perform the one-sample t-test in this scenario:

Step 1: State the hypotheses:
- Null Hypothesis (H0): There is no significant difference between the mean number of anagrams solved by anxiety-ridden participants (μ) and the population mean (µ = 26).
- Alternative Hypothesis (Ha): There is a significant difference, and the mean number of anagrams solved by anxiety-ridden participants (μ) is less than the population mean (µ < 26).

Step 2: Set the significance level (α):
The significance level (α) is a predetermined threshold to determine the level of evidence required to reject the null hypothesis. Let's assume a significance level of α = 0.05, which is a common choice.

Step 3: Compute the test statistic and the p-value:
We will calculate the t-statistic using the given sample data and then find the corresponding p-value.

The formula to calculate the t-statistic for a one-sample t-test is:
t = (x̄ - μ) / (s / √n),
where:
x̄ is the sample mean (23.36),
μ is the population mean (26),
s is the standard deviation of the sample (4),
n is the sample size (14).

Substituting the values, we get:
t = (23.36 - 26) / (4 / √14) ≈ -1.18

Using the t-table or statistical software, we can find the p-value associated with the calculated t-value.

Step 4: Make a decision:
Compare the p-value to the significance level (α) to determine whether to reject or fail to reject the null hypothesis.

If the p-value is less than α (p < α), we reject the null hypothesis and conclude that there is a significant difference between the mean number of anagrams solved by anxiety-ridden participants and the population mean. In this case, it would suggest a decrease in task performance.

If the p-value is greater than or equal to α (p ≥ α), we fail to reject the null hypothesis. It means we do not have enough evidence to conclude that there is a significant difference.

Note: The direction of the alternative hypothesis (less than or greater than) would depend on the research question and the expected outcome.

So, to answer the question, we need the p-value associated with the calculated t-value to make a decision.