Statistics Cheatsheet 3

by Lei Bao

Sampling Distributions

Terms and Definitions

• $\bar{x}$: Sample

  • $\mu_{\bar{x}}$: Mean of the Sampling Distribution of $\bar{x}$

  • $\sigma_{\bar{x}}$: Standard Deviation of the Sampling Distribution of $\bar{x}$

• $\hat{p}$: Sample Proportion

  • $\mu_{\hat{p}}$: Mean of the Sampling Distribution of $\hat{p}$

  • $\sigma_{\hat{p}}$: Standard Deviation of the Sampling Distribution of $\hat{p}$

Typical Problems

1. Suppose a geyser has a mean time between eruptions of 79 minutes. Let the interval of time between the eruptions be normally distributed with standard deviation 30 minutes.

• What is the probability that a random sample of 12 time intervals between eruptions ha a mean longer than 92 minutes?

  • Mean of the Sampling Distribution of $\bar{x}$

  • $\mu_{\bar{x}} = \mu$

  • $\mu_{\bar{x}} = 79$

  • Standard Deviation of the Sampling Distribution of $\bar{x}$

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

  • $\sigma_{\bar{x}} = \frac{30}{\sqrt{12}}=8.6603$

  • Standardizing a Normal Random Variable

  • $z_{\bar{x}} = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}}$

  • $z_{\bar{x}} = \frac{92 - 79}{8.6603}= 1.50$

  • Z table

  • Area in left tail for $z_{\bar{x}}$: 0.9332

  • Area in right tail for $z_{\bar{x}}$: 1 - 0.9332 = 0.0668

• What is the probability that a random sample of 23 time intervals between eruptions has a mean longer than 92 minutes?

  • $\mu_{\bar{x}} = 79$

  • Standard Deviation of the Sampling Distribution of $\bar{x}$

  • $\sigma_{\bar{x}} = \frac{30}{\sqrt{23}}=6.2554$

  • Area in left tail for $z_{\bar{x}}$: 0.9812

  • Area in right tail for $z_{\bar{x}}$: 1 - 0.9812 = 0.0188

• What effect does increasing the sample size have on the probability?

  • If the population mean is less than 92 minutes, then the probability that the sample mean of the time between eruptions is greater than 92 minutes decreases because the variability in the sample mean decreases as the sample size increases.

• What might you conclude if a random sample of 23 time intervals between eruptions has a mean longer than 92 minutes?

  • The population mean may be greater than 79.

  • The population mean is 79, and this is just a rare sampling.