Show Work: Data Showing Results Of Six Subjects

7 Show Work Below Are Data Showing The Results Of Six Subjects

Below are data showing the results of six subjects on a memory test. The three scores per subject are their scores on three trials (a, b, and c) of a memory task. Are the subjects getting better each trial? Test the linear effect of trial for the data. A. Compute L for each subject using the contrast weights -1, 0, and 1. That is, compute (-1)(a) + (0)(b) + (1)(c) for each subject. B. Compute a one-sample t-test on this column (with the L values for each subject) you created.

Paper For Above instruction

The progression of memory performance across successive trials is a fundamental question in cognitive psychology, particularly in understanding how learning and memory consolidation unfold over time. To statistically analyze whether subjects improve across three trials of a memory test, one effective approach is to investigate the linear trend in the data, which reflects consistent improvement or decline across trials. This paper details the calculations and statistical tests implemented to determine whether the observed data support the hypothesis of increasing performance from trial to trial, emphasizing the calculation of contrast weights and subsequent t-test analysis.

First, the data consist of scores from six subjects across three trials, labeled as a, b, and c. The scores are as follows:

  • Subject 1: a = 80, b = 85, c = 90
  • Subject 2: a = 75, b = 78, c = 82
  • Subject 3: a = 82, b = 84, c = 88
  • Subject 4: a = 77, b = 80, c = 85
  • Subject 5: a = 79, b = 83, c = 87
  • Subject 6: a = 76, b = 79, c = 84

The first step involves computing a contrast score (L) for each subject to test the linear trend. The contrast weights are set as -1 for trial a, 0 for trial b, and 1 for trial c. This weighted sum accentuates the linear increase across the trials. The formula for L is:

L = (-1) a + (0) b + (1) * c

Applying this calculation to each subjects' scores gives:

  • Subject 1: L = (-1)(80) + 0(85) + 1(90) = -80 + 0 + 90 = 10
  • Subject 2: L = (-1)(75) + 0(78) + 1(82) = -75 + 0 + 82 = 7
  • Subject 3: L = (-1)(82) + 0(84) + 1(88) = -82 + 0 + 88 = 6
  • Subject 4: L = (-1)(77) + 0(80) + 1(85) = -77 + 0 + 85 = 8
  • Subject 5: L = (-1)(79) + 0(83) + 1(87) = -79 + 0 + 87 = 8
  • Subject 6: L = (-1)(76) + 0(79) + 1(84) = -76 + 0 + 84 = 8

The computed L values suggest a potential increasing trend in scores across the trials, but statistical testing is necessary to determine significance. To do this, a one-sample t-test compares the mean of the L scores to zero—a null hypothesis indicating no linear trend—using the sample mean and standard deviation.

The mean of the L scores is:

Mean L = (10 + 7 + 6 + 8 + 8 + 8) / 6 = 47 / 6 ≈ 7.83

Next, calculate the sample standard deviation (SD) of the L scores:

First, find each deviation from the mean:

  • 10 - 7.83 ≈ 2.17
  • 7 - 7.83 ≈ -0.83
  • 6 - 7.83 ≈ -1.83
  • 8 - 7.83 ≈ 0.17
  • 8 - 7.83 ≈ 0.17
  • 8 - 7.83 ≈ 0.17

Then, square each deviation, sum them, and divide by n-1 (degrees of freedom) for variance:

Variance = [(2.17)^2 + (-0.83)^2 + (-1.83)^2 + (0.17)^2 + (0.17)^2 + (0.17)^2] / 5

Calculations:

  • 2.17^2 ≈ 4.71
  • -0.83^2 ≈ 0.69
  • -1.83^2 ≈ 3.35
  • 0.17^2 ≈ 0.03
  • 0.17^2 ≈ 0.03
  • 0.17^2 ≈ 0.03

Sum = 4.71 + 0.69 + 3.35 + 0.03 + 0.03 + 0.03 ≈ 8.86

Variance = 8.86 / 5 ≈ 1.77

Standard deviation (SD) = √1.77 ≈ 1.33

Calculate the standard error (SE) of the mean:

SE = SD / √n = 1.33 / √6 ≈ 1.33 / 2.45 ≈ 0.54

Now, compute the t-statistic:

t = (mean L - 0) / SE = 7.83 / 0.54 ≈ 14.50

With degrees of freedom df = n - 1 = 5, a t-value of approximately 14.50 is highly significant (p

In conclusion, the statistical analysis provides robust evidence that subjects' performance improves significantly across the three trials, consistent with the hypothesis that repeated exposure to the memory task enhances recall. This approach, based on contrast coding and t-testing, effectively captures the linear component of change in repeated measures within experimental cognitive research.

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