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Why the sample variance is an unbiased estimator of the population variance?
The sample variance is considered an unbiased estimator of the population variance because it corrects for the bias introduced by estimating the population variance from a sample. When calculating the sample variance, we use ( n-1 ) (where ( n ) is the sample size) instead of ( n ) in the denominator, which compensates for the degree of freedom lost when estimating the population mean from the sample. This adjustment ensures that the expected value of the sample variance equals the true population variance, making it an unbiased estimator.
What does it means if the standard deviation is large?
that you have a large variance in the population and/or your sample size is too small
What is n-1in statistics?
In statistics, "n-1" refers to the degrees of freedom used in the calculation of sample variance and sample standard deviation. When estimating variance from a sample rather than a whole population, we divide by n-1 (the sample size minus one) instead of n to account for the fact that we are using a sample to estimate a population parameter. This adjustment corrects for bias, making the sample variance an unbiased estimator of the population variance. It is known as Bessel's correction.
What is a fundamental difference between the t statistic and a z score?
The fundamental difference between the t statistic and a z score lies in the sample size and the underlying population variance. The t statistic is used when the sample size is small (typically n < 30) and the population variance is unknown, making it more appropriate for estimating the mean of a normally distributed population. In contrast, the z score is used when the sample size is large or when the population variance is known, as it assumes a normal distribution of the sample mean. Consequently, the t distribution is wider and has heavier tails than the z distribution, reflecting greater uncertainty in smaller samples.
What is not dependent on the size of a sample?
In general the mean of a truly random sample is not dependent on the size of a sample. By inference, then, so is the variance and the standard deviation.
Why the sample variance is an unbiased estimator of the population variance?
The sample variance is considered an unbiased estimator of the population variance because it corrects for the bias introduced by estimating the population variance from a sample. When calculating the sample variance, we use ( n-1 ) (where ( n ) is the sample size) instead of ( n ) in the denominator, which compensates for the degree of freedom lost when estimating the population mean from the sample. This adjustment ensures that the expected value of the sample variance equals the true population variance, making it an unbiased estimator.
How much error between sample mean and population mean?
The answer depends on the underlying variance (standard deviation) in the population, the size of the sample and the procedure used to select the sample.
What does it means if the standard deviation is large?
that you have a large variance in the population and/or your sample size is too small
What is n-1in statistics?
In statistics, "n-1" refers to the degrees of freedom used in the calculation of sample variance and sample standard deviation. When estimating variance from a sample rather than a whole population, we divide by n-1 (the sample size minus one) instead of n to account for the fact that we are using a sample to estimate a population parameter. This adjustment corrects for bias, making the sample variance an unbiased estimator of the population variance. It is known as Bessel's correction.
If a sample of size 3 students showed the following grades in a certain exam 74 75 and 76 then the variance is?
The sample variance is 1.
What is a fundamental difference between the t statistic and a z score?
The fundamental difference between the t statistic and a z score lies in the sample size and the underlying population variance. The t statistic is used when the sample size is small (typically n < 30) and the population variance is unknown, making it more appropriate for estimating the mean of a normally distributed population. In contrast, the z score is used when the sample size is large or when the population variance is known, as it assumes a normal distribution of the sample mean. Consequently, the t distribution is wider and has heavier tails than the z distribution, reflecting greater uncertainty in smaller samples.
What is not dependent on the size of a sample?
In general the mean of a truly random sample is not dependent on the size of a sample. By inference, then, so is the variance and the standard deviation.
If the sample mean is 10 the hypothesized population mean is 9 and the population standard deviation is 4 compute the test value needed for the z test?
n = sample sizen1 = sample 1 sizen2 = sample 2 size= sample meanμ0 = hypothesized population meanμ1 = population 1 meanμ2 = population 2 meanσ = population standard deviationσ2 = population variance
What is the sample variance and the estimated standard error for a sample of n 9 and scores with SS 72?
The sample variance (s²) is calculated using the formula ( s² = \frac{SS}{n - 1} ), where SS is the sum of squares and n is the sample size. For a sample size of n = 9 and SS = 72, the sample variance is ( s² = \frac{72}{9 - 1} = \frac{72}{8} = 9 ). The estimated standard error (SE) is the square root of the sample variance divided by the sample size, calculated as ( SE = \sqrt{\frac{s²}{n}} = \sqrt{\frac{9}{9}} = 1 ). Thus, the sample variance is 9 and the estimated standard error is 1.
Will a large sample size and a small sample variance produce the largest value for the estimated standard error?
no
As the sample size increases what does the expected value of M do?
You have not defined M, but I will consider it is a statistic of the sample. For an random sample, the expected value of a statistic, will be a closer approximation to the parameter value of the population as the sample size increases. In more mathematical language, the measures of dispersion (standard deviation or variance) from the calculated statistic are expected to decrease as the sample size increases.
In an analysis of variance which is directly influenced by the size of the sample variances?
df within
