Sampling Distributions¶

Sampling Distributions for Sample Means¶

Take all possible samples of size \(n\) from a population and calculate the sample mean \(\overline{x}\) for each, all possible values of the sample mean are calculated

Parameters of Sampling Distributions for Sample Means¶

• Mean = \(\mu\)
• Standard deviation = \(\dfrac{\sigma}{\sqrt{n}}\)
• Standardized \(z\)-score = \(\dfrac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\)

Normality of Sampling Distributions for Sample Means¶

• If the population is normally distributed, the sampling distribution for sample means is normally distributed

Central Limit Theorem¶

• If a population is not normally distributed
• A large enough random sample of size \(n \geq 30\) is taken while sample values are independent
• Then the sampling distribution for sample means is approximately normally distributed

Sampling Distributions for Differences in Sample Means¶

One-Sample Problem¶

When one random sample of size \(n\) has been taken from one population

Two-Sample Problem¶

If one random sample of size \(n_1\) is taken from one population, then a different random sample of size \(n_2\) is taken from a different population that is independent to the first population

Parameters of Sampling Distribution for Differences in Sample Means¶

• Mean = \(\mu_1 - \mu_2\)
• Standard deviation = \(\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}\)
  • Sampling with replacement or each sample size is less than 10% of the population size
  • Otherwise, \(\sigma\) will be smaller
• Standardized \(z\)-score = \(\dfrac{(\overline{x_1} - \overline{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}}\)

Normality of Sampling Distributions for Differences in Sample Means¶

• If two independent populations are normally distributed, the sampling distribution for differences in sample means is also normally distributed

Sampling Distributions for Sample Proportions¶

• Population proportion, \(p\), is the percentage of success
• \(n\) is the sample size
• \(X\) is the number of successes in a sample, following binomial distribution
• Sample proportion \(\hat{p} = \dfrac{X}{n}\)

If

• \(np \geq 10\)
• \(n(1-p) \geq 10\)

Then \(\hat{p}\)

• Normally distributed
• Mean = \(p\)
• Standard deviation = \(\sqrt{\dfrac{p(1-p)}n}\)
• \(z\)-score = \(\dfrac{\hat{p} - p}{{\sqrt{\dfrac{(1-p)}{n}}}}\)

Sampling Distributions for Differences in Sample Proportions¶

If

• \(n_1p_1 \geq 10\)
• \(n_1(1-p_1) \geq 10\)
• \(n_2p_2 \geq 10\)
• \(n_2(1-p_2) \geq 10\)

Then \(\hat{p_1} - \hat{p_2}\)

• Normally distributed
• Mean = \(p_1 - p-2\)
• Standard deviation = \(\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}\)
• Standardized \(z\)-score = \(\dfrac{(\hat{p_1} - \hat{p_2}) - (p_1 - p_2)}{\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}}\)

Biased & Unbiased Estimators¶

Estimator is used to estimate the population parameter

To know if an estimator is a good predictor

• All possible estimates from all possible samples of size \(n\) must be generated
• Check to see if, on average, estimates are centered around the value of the population parameter
• An estimator is said to be unbiased if the mean of its sampling distribution equals the population parameter being estimated

Unbiased Estimators¶

• Sample mean \(\overline{x}\)
• Sample proportion \(\hat{p}\)
• Sample standard deviation \(s\)

Affects on Unbiased Estimators¶

• \(n\) does not affect \(\overline{x}\)
• Greater \(n\) gives more accurate \(S = \dfrac{\sigma}{\sqrt{n}}\)