Considered the sampling distribution of a sample mean obtained by random sampling from an infinite population. This population has a distribution that is highly skewed toward the larger values.

a) How is the mean of the sampling distribution related to the mean of the population?

b) How is the standard deviation of the sampling distribution related to the standard deviation of the population?

c) How is the shape if the sampling distribution affected by the sample size?

*please explain the analyses to the answer of each question*

The mean of the sampling distribution of the mean is always equal to the population mean. The standard deviation of the sampling distribution of the mean is called the standard error of the mean and represents the average deviation of the sample means from the population mean. The shape of the sampling distribution approaches a normal distribution as sample size increases.

Evaluate the following expressions for the given set of scores:

a. ΣX  b. ΣX2  c. (ΣX)2  d. Σ2X  e. Σ(X – 2)
Scores:
                                 X
                                      0
                                      5
                                      3
                                      2

a) The mean of the sampling distribution is equal to the mean of the population. This means that, on average, the sample means will be the same as the population mean. This is true regardless of the shape of the population distribution.

b) The standard deviation of the sampling distribution, also known as the standard error of the mean, is related to the standard deviation of the population by the formula: standard error of the mean = population standard deviation / square root of sample size. In other words, the standard deviation of the sampling distribution decreases as the sample size increases. This means that as more samples are taken, the sample means are more likely to cluster around the population mean.

c) The shape of the sampling distribution is affected by the sample size. When the sample size is small, the sampling distribution may resemble the shape of the population distribution, which in this case would be highly skewed toward larger values. However, as the sample size increases, the shape of the sampling distribution approaches a normal distribution. This is known as the Central Limit Theorem. In practical terms, this means that even if the population distribution is highly skewed, the sampling distribution of the mean will become more symmetrical and bell-shaped as sample size increases.

a) The mean of the sampling distribution is equal to the mean of the population. This means that if we were to take many random samples from the population and calculate the mean for each sample, the average of those sample means would be equal to the population mean. This relationship holds regardless of the shape of the population distribution.

To understand this relationship, we can consider the concept of unbiasedness. If our random sampling process is unbiased, meaning that each member of the population has an equal chance of being selected, then the sample means will, on average, be representative of the population mean. This is why the mean of the sampling distribution is equal to the population mean.

b) The standard deviation of the sampling distribution of the mean, also known as the standard error of the mean, is related to the standard deviation of the population. It can be calculated using the formula:

standard error of the mean = standard deviation of the population / square root of sample size

As you can see, the standard error of the mean is inversely proportional to the square root of the sample size. This means that as the sample size increases, the standard error decreases. In other words, larger sample sizes tend to produce more reliable or precise estimates of the population mean.

The reason for this relationship is that when we take larger samples, we are more likely to include a diverse range of observations from the population. This tends to reduce the variability in the sample means, resulting in a smaller standard error.

c) The shape of the sampling distribution is affected by the sample size. When the population distribution is highly skewed towards larger values, the sampling distribution of the mean typically becomes less skewed as the sample size increases. As the sample size increases, the Central Limit Theorem comes into play, which states that the distribution of the sample means will tend to be approximately normal, regardless of the shape of the population distribution, as long as the sample size is sufficiently large.

In simpler terms, when we take larger samples, the sample means tend to be less influenced by extreme values or outliers in the population. This leads to a more symmetric and bell-shaped sampling distribution. On the other hand, with smaller sample sizes, the sampling distribution may still retain some of the skewness from the population distribution.

Overall, the shape of the sampling distribution becomes more normal as the sample size increases, allowing us to make more reliable inferences about the population mean.