Describe the sampling distribution of p(hat). Assume the size of the population is 30,000.
n=800, p=0.6
a) Determine the mean of the sampling distribution
b) Dtermine the standard deviation of the sampling distribution
n=800, p=0.6
a) Determine the mean of the sampling distribution
b) Dtermine the standard deviation of the sampling distribution
In the scenarios you mentioned, where the population is normally distributed, skewed, and uniform, the impact on the population mean and standard deviation will vary.
1. Normally Distributed Population:
When the population is normally distributed, the sample mean will be an unbiased estimator of the population mean, regardless of the sample size or the number of samples. This means that, on average, the sample mean should be equal to the population mean. As the sample size increases, the sample mean will tend to have less variability or fluctuations around the population mean. The standard deviation of the sample mean (also known as standard error) will decrease as the sample size increases, indicating a higher precision in estimating the population mean.
2. Skewed Population:
If the population is skewed, the sample mean may still be a reasonable estimator of the population mean, but it may not be as accurate as in the normally distributed population case. Skewed populations have asymmetric distributions, where the tail on one side is longer or heavier than the other. In such cases, the sample mean may be influenced more by extreme values in the population, leading to some bias or shifts away from the true population mean. The standard deviation of the sample mean will still decrease with larger sample sizes, but it may not decrease as rapidly compared to normally distributed populations.
3. Uniformly Distributed Population:
In a uniformly distributed population, where all values have equal probabilities of occurring, the mean of the population is simply the midpoint of the range. As for the standard deviation, it depends on the range of the distribution. If you increase the sample size, the sample mean will have less variation and will be a better estimator of the population mean. However, the standard deviation will not necessarily decrease or increase uniformly depending on the number of samples or sample size.
It's important to note that these are general observations about the impact of sample size, number of samples, and population distribution, but actual results can vary depending on other factors and assumptions.