Question

4. A study of the differences in cognitive function between normal individuals and patients diagnosed with schizophrenia. The total time (in minutes) a subject spent on the Trail Making Test was used as a measure of cognitive function. Summary data for independent random samples of 41 schizophrenics and 49 normal individuals are reproduced below. Suppose the researcher theorize that schizophrenics will have a wider range in time on the Trail Making Test than will normal subjects. Is there evidence to support this theory? Test using α-0.05. Schizophrenia Normal 49 62.24 16.34 Sample size Mean time 41 104.23 Standard deviation: 445

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis H₀: σ_A²< σ_B²

Alternative hypothesis H_A: σ_A² > σ_B²

Formulate an analysis plan. For this analysis, the significance level is 0.05.

Analyze sample data. Using sample data, the degrees of freedom (DF), and the test statistic (F).

DF₁ = n₁ - 1 = 41 -1

D.F₁ = 40

DF₂ = n₂ - 1 = 49 -1

D.F₂ = 48

Test statistics:-

$F = \frac{s_{1}^{2}}{s_{2}^{2}}$

F = 7.737

Since the first sample had the larger standard deviation, this is a right-tailed test.

p value for the F distribution = less than 0.001

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favour of the claim that schiophrenics will have a wider range in time o the trial making test than will normal subjects.