Answer: 1.To find the mean, we sum up all the scores and divide by the number of scores:
Mean = (5 + 12 + 8 + 14 + 11) / 5 = 50 / 5 = 10
To find the variance, we first find the deviation of each score from the mean:
5 - 10 = -5
12 - 10 = 2
8 - 10 = -2
14 - 10 = 4
11 - 10 = 1
We square each deviation and sum them up:
(-5)^2 + 2^2 + (-2)^2 + 4^2 + 1^2 = 26
Then, we divide by the number of scores to find the variance:
Variance = 26 / 5 = 5.2
Finally, we find the standard deviation by taking the square root of the variance:
Standard deviation = √5.2 = 2.28
2. To construct a simple random sample of size 2, we can choose any two scores from the population. The number of ways to do this is given by the combination formula:
C(5,2) = 5! / (2! * (5 - 2)!) = 10
Therefore, there are 10 simple random samples of size 2 that can be constructed.
3.The mean of the sampling distribution of the sample means is equal to the population mean, which we found to be 10. The variance of the sampling distribution is equal to the population variance divided by the sample size:
Variance of sampling distribution = Variance / Sample size = 5.2 / 2 = 2.6
The standard error of the sampling distribution is the square root of the variance:
Standard error = √2.6 = 1.61
4.The variance of the sampling distribution is 2.6, as we found in part
5. Since this is considered to be data from a simple random sample, we can use the sample mean as a point estimate of the population mean. Therefore, the point estimate of the population mean is 10.
Similarly, we can use the sample variance as a point estimate of the population variance. However, we need to correct for the bias in the sample variance by dividing by (n-1) instead of n:
Point estimate of population variance = Variance * (n / (n-1)) = 5.2 * (5 / 4) = 6.5
Explanation: