Exam Notes on Population and Sampling Distribution

Population

• Definition: The entire group of individuals or items that are of interest.

Sampling Distribution

• Definition: The distribution of a statistic (e.g., sample mean, sample proportion) calculated from multiple samples of the same size, drawn from the same population.

Variable

• Definition: A characteristic or attribute that can take on different values.

• Types:

  • Categorical: Variables that represent categories or groups (e.g., left-handed or not).

  • Quantitative: Variables that represent numerical values (e.g., age).

Parameter

• Definition: A numerical value that describes a characteristic of a population.

• Example:

  • $p$ = population proportion

  • $\mu$ = population mean

  • $\sigma$ = population standard deviation

Sample

• Definition: A subset of the population.

Statistic

• Definition: A numerical value that describes a characteristic of a sample.

• Example:

  • $\hat{p}$ = sample proportion

  • $\bar{x}$ = sample mean

Center, Spread, Shape

• These are used to describe distributions.

Normal Distribution of Sample Mean

• The distribution of sample means will be approximately normal IF $np \geq 10$ and $n(1-p) \geq 10$.

Distribution of Sample Mean

• Denoted as $N(\mu, \frac{\sigma}{\sqrt{n}})$

• The mean of the distribution of sample means ($\mu<em>{\bar{x}}$) is equal to the population mean ($\mu$): $\mu</em>{\bar{x}} = \mu$

Central Limit Theorem (CLT)

• If samples of size $n \geq 30$ are drawn from any population with mean $\mu$ and standard deviation $\sigma$, then the sampling distribution of the sample means approximates a normal distribution.

• The greater the sample size, the better the approximation.

Sampling Distribution of the Mean

• The distribution of sample means, with all samples having the same sample size $n$ taken from the same population.

• Typically represented as a probability distribution in the format of a table, probability histogram, or formula.

Z-value for Sampling Distribution of the Mean

• Formula: $Z = \frac{(\bar{X} - \mu)}{(\frac{\sigma}{\sqrt{n}})}$

  • $\bar{X}$ = sample mean

  • $\mu$ = population mean

  • $\sigma$ = population standard deviation

  • $n$ = sample size

Example Using Central Limit Theorem

• The population mean annual salary for plumbers is $46,700. A random sample of 42 plumbers is drawn from this population. What is the probability that the mean salary of the sample is greater than $44,000? Assume \sigma = $5600.

Example: Preventative Maintenance on Air Conditioners

• The time (in hours) that a technician requires to complete preventative maintenance on an air conditioner follows a distribution that is strongly right-skewed with a mean of 1 hour and a standard deviation of 0.1.

• If your company will service an SRS of 70 air conditioners and you have budgeted 1.1 hours per unit, will this be enough?

• The sampling distribution of the mean time spent working is approximately $N(1, \frac{0.1}{\sqrt{70}})$

  • P(\bar{X} > 1.1) = P(Z > \frac{1.1 - 1}{0.1/\sqrt{70}}) = P(Z > 8.37) \approx 0

Example: Sampling Distribution of Sample Means

• Find the mean, variance, and standard deviation of the sampling distribution of the sample means.

• $\sigma = 3$ and $n=5$

• $\mu_{\bar{x}} = \mu$

• $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{3}{\sqrt{5}} = 1.3416$

Distribution of Sample Means (N-5)

• Mean = 3.0

• SD = 0.63

• For n = 1

• y = 0-20

Statistical Error

• Suppose we need to measure the length of a line. Using 5 rulers and taken at different times we get the following lengths: 971, 972, 973, 979 and 980 cm.

• How best to get the ‘best’ estimate of the length.

  • Use the median 973

  • Use the mean 975

Why is the mean a better measure?

• It minimizes the potential error of the estimate.

MSE (Mean Squared Error)

• Comparison of Mean Sq. Errors (Btw Value and Original Data)

Standardized Normal Distribution

• Formula: $Z = \frac{X - \mu}{\sigma}$