sample mean

Examples of sample mean in the following topics:

• Working Backwards to Find the Error Bound or Sample Mean

  • Working Backwards to find the Error Bound or the Sample Mean
  • From the upper value for the interval, subtract the sample mean
  • We may know that the sample mean is 68.
  • If we know that the sample mean is 68: EBM = 68.82 − 68 = 0.82
  • If we don't know the sample mean: EBM = (68.82 − 67.18)/2 = 0.82

• Mean of All Sample Means (μ x)

  • The mean of the distribution of differences between sample means is equal to the difference between population means.
  • Compute the means of the two samples ( M1 and M2);
  • which says that the mean of the distribution of differences between sample means is equal to the difference between population means.
  • If numerous samples were taken from each age group and the mean difference computed each time, the mean of these numerous differences between sample means would be 34 - 25 = 9.
  • The variance sum law states that the variance of the sampling distribution of the difference between means is equal to the variance of the sampling distribution of the mean for Population 1 plus the variance of the sampling distribution of the mean for Population 2.

• Which Standard Deviation (SE)?

  • For example, the sample mean is the usual estimator of a population mean.
  • However, different samples drawn from that same population would in general have different values of the sample mean.
  • The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population.
  • However, the mean and standard deviation are descriptive statistics, whereas the mean and standard error describes bounds on a random sampling process.
  • Put simply, standard error is an estimate of how close to the population mean your sample mean is likely to be, whereas standard deviation is the degree to which individuals within the sample differ from the sample mean.

• Sampling Distributions and the Central Limit Theorem

  • The central limit theorem for sample means states that as larger samples are drawn, the sample means form their own normal distribution.
  • The central limit theorem for sample means specifically says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution).
  • The upshot is that the sampling distribution of the mean approaches a normal distribution as $n$, the sample size, increases.
  • The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution.
  • It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case).

• Properties of Sampling Distributions

  • For example, knowing the degree to which means from different samples differ from each other and from the population mean would give you a sense of how close your particular sample mean is likely to be to the population mean.
  • The most common measure of how much sample means differ from each other is the standard deviation of the sampling distribution of the mean.
  • If all the sample means were very close to the population mean, then the standard error of the mean would be small.
  • On the other hand, if the sample means varied considerably, then the standard error of the mean would be large.
  • To be specific, assume your sample mean is 125 and you estimated that the standard error of the mean is 5.

• Introduction to one-sample means with the t distribution

  • The motivation in Chapter 4 for requiring a large sample was two-fold.
  • First, a large sample ensures that the sampling distribution of $\bar{x}$ is nearly normal.
  • The second motivation for a large sample was that we get a better estimate of the standard error when using a large sample.
  • We will see that the t distribution is a helpful substitute for the normal distribution when we model a sample mean $\bar{x}$ that comes from a small sample.
  • While we emphasize the use of the t distribution for small samples, this distribution may also be used for means from large samples.

• Standard Error

  • For example, the sample mean is the usual estimator of a population mean.
  • The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population.
  • As mentioned, the standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean.
  • It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator.
  • Put simply, standard error is an estimate of how close to the population mean your sample mean is likely to be, whereas standard deviation is the degree to which individuals within the sample differ from the sample mean.

• What Is a Sampling Distribution?

  • You would not expect your sample mean to be equal to the mean of all women in Houston.
  • Similarly, if you took a second sample of 10 women from the same population, you would not expect the mean of this second sample to equal the mean of the first sample.
  • This statistic is then called the sample mean.
  • An alternative to the sample mean is the sample median.
  • You would not expect your sample mean to be equal to the mean of all women in Houston.

• Introduction to Sampling Distributions

  • You would not expect your sample mean to be equal to the mean of all women in Houston.
  • Similarly, if you took a second sample of 10 people from the same population, you would not expect the mean of this second sample to equal the mean of the first sample.
  • Specifically, it is the sampling distribution of the mean for a sample size of 2 (N = 2).
  • The most common measure of how much sample means differ from each other is the standard deviation of the sampling distribution of the mean.
  • If all the sample means were very close to the population mean, then the standard error of the mean would be small.

• t-Test for Two Samples: Independent and Overlapping

  • Two-sample t-tests for a difference in mean involve independent samples, paired samples, and overlapping samples.
  • The two sample t-test is used to compare the means of two independent samples.
  • For the null hypothesis, the observed t-statistic is equal to the difference between the two sample means divided by the standard error of the difference between the sample means.
  • Comparing means based on convenience sampling or non-random allocation is meaningless.
  • Two-sample t-tests for a difference in mean involve independent samples, paired samples and overlapping samples.