Steven collected data from 20 college students on their emotional responses to classical music. Students listened to two 30-second segments from “The Collection from the Best of Classical Music.” After listening to a segment, the students rated it on a scale from 1 to 10, with 1 indicating that it “made them very sad” to 10 indicating that it “made them very happy.” Steve computes the total scores from each student and created a variable called “hapsad.” Steve then conducts a one-sample t-test on the data, knowing that there is an established mean for the publication of others that have taken this test of 6. The following is the scores:

5.0 5.0
10.0 3.0
13.0 13.0
7.0 5.0
5.0 15.0
14.0 18.0
8.0 12.0
10.0 7.0
3.0 15.0
4.0 3.0

Conduct a one-sample t-test. What is the t-test score? What is the mean? Was the test significant? If it was significant at what P-value level was it significant?
What is your null and alternative hypothesis? Given the results did you reject or fail to reject the null and why?
(Use instructions on page 437 of your textbook, under Hypothesis Tests with the t Distribution to conduct SPSS or Excel analysis).

To conduct a one-sample t-test, you can follow these steps:

Step 1: Write the null and alternative hypotheses.
The null hypothesis (H0) states that there is no significant difference between the mean emotional response scores and the established mean of 6: μ = 6.
The alternative hypothesis (Ha) states that there is a significant difference between the mean emotional response scores and the established mean: μ ≠ 6.

Step 2: Calculate the mean of the sample.
To calculate the mean, sum up all the emotional response scores and divide by the number of observations (in this case, 20).

Mean = (5.0 + 5.0 + 10.0 + 3.0 + 13.0 + 13.0 + 7.0 + 5.0 + 14.0 + 18.0 + 8.0 + 12.0 + 10.0 + 7.0 + 3.0 + 4.0 + 3.0) / 20

Mean = 168 / 20

Mean = 8.4

Step 3: Calculate the standard deviation of the sample.
To calculate the standard deviation, you will need to find the differences between each observation and the mean, square those differences, sum them up, divide by the number of observations, and finally take the square root.

SD = √[(5.0 - 8.4)² + (5.0 - 8.4)² + ... + (4.0 - 8.4)²] / 20

SD = √[(-3.4)² + (-3.4)² + ... + (-4.4)²] / 20

SD = √[11.56 + 11.56 + ... + 19.36] / 20

SD = √[203.84] / 20

SD = √10.192 / 20

SD = 0.498

Step 4: Calculate the t-test score.
The t-test score is calculated by dividing the difference between the sample mean and the expected mean (μ - μ0) by the standard deviation divided by the square root of the sample size (SD / √n).

t = (Mean - μ0) / (SD / √n)

t = (8.4 - 6) / (0.498 / √20)

t = 2.4 / (0.498 / √20)

t = 2.4 / (0.498 / 4.472)

t = 2.4 / (0.1114)

t ≈ 21.53 (Note: rounded to 2 decimal places)

Step 5: Determine the degrees of freedom.
The degrees of freedom (df) for a one-sample t-test is equal to the sample size minus 1.

df = n - 1

df = 20 - 1

df = 19

Step 6: Determine the significance level and critical t-value.
The significance level (α) represents the level of risk for rejecting the null hypothesis. In this case, we'll assume a significance level of 0.05.
The critical t-value is obtained from a t-distribution table using the degrees of freedom and the significance level.

For a two-tailed test with α = 0.05 and 19 degrees of freedom, the critical t-value is approximately ±2.093.

Step 7: Conduct the test and interpret the results.
To interpret the results, we compare the calculated t-test score (21.53) with the critical t-value (±2.093).

Since the calculated t-test score is much larger than the critical t-value (21.53 > ±2.093), the test is significant.

Given the results, we reject the null hypothesis (H0) and conclude that there is a significant difference between the mean emotional response scores and the established mean of 6. The emotional response to classical music, as measured by the "hapsad" variable, differs significantly from the established mean.