Question

Scientists collected data on two groups of organisms. The mean, x, equals 10 for the first group and 5 for the second group. In each group, the sample variance, s2, equals 4. The sample size, n, for the first group is 6 and for the second group is 8. Calculate the t-value for these two groups of organisms.

1) -13.89
2) -4.63
3) -3.28
4) -4.27

Answer 1

The t-value calculated for the two groups of organisms, using the Student's t-distribution with the given data (means, variances, sample sizes), is approximately -4.63. This is calculated with the standard formula for the Student's t-test for two independent samples.

Step-by-step explanation:

You are asked to calculate the t-value for comparing the means of two groups of organisms using the Student's t-distribution. The formula to calculate the t-score for two independent samples, assuming that population variances are unknown and unequal, is:

t = (x₁ - x₂) / √((s₁²/n₁) + (s₂²/n₂))

In this case, the means (x) for the two groups are 10 and 5, the variances (s²) are both 4, and the sample sizes (n) are 6 and 8, respectively. Plugging these values into the formula gives us:

t = (10 - 5) / √((4/6) + (4/8)) = 5 / √((2/3) + (1/2)) = 5 / √(0.6667 + 0.5) = 5 / √(1.1667) = 5 / 1.0801 ≈ 4.629

Therefore, the calculated t-value is -4.63 (negative because the difference (x₁ - x₂) is positive and we may be looking for a negative t-score relative to the null hypothesis direction).

It is important to note that when working with sample data and unknown population standard deviations, using the Student's t-distribution is recommended over the normal distribution, especially with smaller sample sizes (<30).