Question

A random sample is to be selected from a population that has a proportion of successes \(p=0.65\). Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) d. \(n=50\) b. \(n=20\) e. \(n=100\) c. \(n=30\) f. \(n=200\)

Short answer

For all sample sizes, the mean of the sampling distribution is \(0.65\). For standard deviation, when \(n=10\), it's \(\sqrt{0.65*(1-0.65)/10}\); for \(n=20\), it's \(\sqrt{0.65*(1-0.65)/20}\); for \(n=30\), it's \(\sqrt{0.65*(1-0.65)/30}\); for \(n=50\), it's \(\sqrt{0.65*(1-0.65)/50}\); for \(n=100\), it's \(\sqrt{0.65*(1-0.65)/100}\); for \(n=200\), it's \(\sqrt{0.65*(1-0.65)/200}\).

Step-by-step solution

Calculate the Mean for Each Given Sample Size

The mean of the sampling distribution is equal to the population proportion \(p\). Since the problem stated that \(p = 0.65\), then the mean of the sampling distribution for all sample sizes is \(0.65\).

Calculate the Standard Deviation for Each Given Sample Size

Firstly, input population proportion \(p=0.65\), and then use the formula of standard deviation \(\sqrt{p(1-p)/n}\) to calculate the standard deviation for each sample size. \n a. For \(n=10\), the standard deviation is \(\sqrt{0.65*(1-0.65)/10}\) \n b. For \(n=20\), the standard deviation is \(\sqrt{0.65*(1-0.65)/20}\) \n c. For \(n=30\), the standard deviation is \(\sqrt{0.65*(1-0.65)/30}\) \n d. For \(n=50\), the standard deviation is \(\sqrt{0.65*(1-0.65)/50}\) \n e. For \(n=100\), the standard deviation is \(\sqrt{0.65*(1-0.65)/100}\) \n f. For \(n=200\), the standard deviation is \(\sqrt{0.65*(1-0.65)/200}\).

Conclusion

Thus, for each sample size, you have both mean and standard deviation of the sampling distribution. When the sample size changes, the mean remains the same, but the standard deviation changes. The larger the sample size, the smaller the standard deviation and the closer the sample proportion is to the population proportion.

Key concepts

Population Proportion

The population proportion, often represented as \(p\), is a measure of the occurrence of a particular trait (success, attribute, or outcome) in a population. For example, if we are interested in the proportion of people in a city who are left-handed, \(p\) would represent the ratio of left-handed individuals to the total population of the city. When statisticians refer to 'success', it means any outcome they are tracking.

Understanding the population proportion is essential as it serves as a benchmark for comparing sample data. In the exercise provided, \(p=0.65\) indicates that 65% of the population possesses the trait of interest. This proportion is critical for determining the behavior of sampling distributions, which in turn helps statisticians to infer about the population using sample data.

Standard Deviation

The standard deviation is a measure that quantifies the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of sampling distributions, the standard deviation of the sample proportion (usually denoted as \(\sigma_\hat{p}\)) provides insights into the variability of \(\hat{p}\) across different samples of the same size. It is calculated using the formula \(\sqrt{p(1-p)/n}\), where \(p\) is the population proportion and \(n\) is the sample size. For instance, with a population proportion of 0.65 and a sample size of 20, the standard deviation would be \(\sqrt{0.65*(1-0.65)/20}\).

This value represents how much the sample proportions are expected to vary from one sample to another. Thus, when interpreting the results, it is a crucial factor in assessing the precision of the sample proportion as an estimate of the population proportion.

Sample Size

Sample size, denoted as \(n\), is the number of observations included in a statistical sample. The choice of sample size is a compromise between cost and accuracy: a larger sample size provides more reliable data but requires more resources to collect.

In sampling distributions, the sample size plays a significant role in determining the standard deviation of the sampling distribution of sample proportion \(\hat{p}\). The formula \(\sqrt{p(1-p)/n}\) highlights that as \(n\) increases, the standard deviation decreases. Therefore, larger samples tend to produce more precise estimates of the population proportion.

The exercise shows how varying the sample size, while keeping the population proportion fixed, affects the standard deviation of \(\hat{p}\). For example, a sample size of 10 leads to more variability in \(\hat{p}\) compared to a sample size of 100. This is an essential consideration when designing studies or experiments since deciding on the right sample size affects the quality of your statistical conclusions.

Mean of Sampling Distribution

The mean of a sampling distribution is the average value of the sample means, which can also be referred to as the expected value. For sample proportions, this mean is equal to the population proportion \(p\). Regardless of the sample size, if the random variable \(\hat{p}\) represents the sample proportion, the mean of the sampling distribution of \(\hat{p}\) (denoted as \(\mu_\hat{p}\)) is equal to \(p\).

This consistency of the mean in the sampling distribution assures that if multiple samples are taken and their proportions calculated, the average of these sample proportions will tend to be around the actual population proportion. In our exercise example, the mean of the sampling distribution is 0.65 for every sample size. It demonstrates a key principle called the 'law of large numbers', which states that as the sample size grows, the sample mean gets closer to the population mean.

Thus, the mean of the sampling distribution is critical for understanding the central tendency of the sample proportions and for making inferences about the population.