Final answer:
The estimated standard error of the sample mean difference when the variances are 10 and 6 with each sample size of 4 is calculated to be 2. The distribution of the sample mean can be approximated by the Student's t distribution.
Step-by-step explanation:
To estimate the standard error of the sample mean difference (SEdiff), we use the standard deviations of the two independent samples and their respective sample sizes. Given that the variance for the first sample is 10 and the second sample is 6, and each sample has n = 4 scores, the formula for SEdiff is:
SEdiff = √[(s1² / n1) + (s2² / n2)]
First, calculate the standard deviation for each sample by taking the square root of the variance:
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- Standard deviation of the first sample (s1): √10 = 3.16
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- Standard deviation of the second sample (s2): √6 = 2.45
Then plug these values into the formula:
SEdiff = √[(3.16² / 4) + (2.45² / 4)] = √[(10 / 4) + (6 / 4)] = √[2.5 + 1.5] = √4 = 2
Therefore, the standard error of the sample mean difference, SEdiff, is 2, rounded to two decimal places.
Regarding the distribution of the sample mean, if both sample sizes are equal or larger than five, the distribution can often be approximated using the Student's t distribution, especially when using estimate standard deviations from samples and not the population standard deviation. This is the basis of forming a t-score for hypothesis testing.