Normal Distributions, Confidence Intervals, and Sampling Distributions: Study Notes

Normal Probability Distributions

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is denoted as N(0, 1). Many problems involving normal distributions can be solved by converting values to z-scores and using the standard normal table.

• Z-score: The number of standard deviations a value is from the mean.

• Formula: $z = \frac{x - \mu}{\sigma}$

• Application: Z-scores allow comparison of values from different normal distributions.

Finding Areas and Probabilities

To find the probability that a value falls within a certain range in a normal distribution, convert the value(s) to z-scores and use the standard normal table.

• Left-tail probability: $P(X < x)$

• Right-tail probability: $P(X > x) = 1 - P(X < x)$

• Between two values: $P(a < X < b) = P(X < b) - P(X < a)$

Example: Find $P(X < 1.13)$ for $N(0,1)$. Convert to z-score (already standard), then use the table: $P(Z < 1.13) = 0.8708$.

Finding Values from Probabilities

Given a probability (area), you can find the corresponding z-score using the standard normal table, then convert back to the original scale if needed.

• Example: Find the z-score with 75% of the distribution to the left: $z = 0.674$.

• For a normal distribution with mean $\mu$ and standard deviation $\sigma$: $x = \mu + z\sigma$

Applications of the Normal Distribution

Real-World Example: ACT Scores

Suppose ACT scores are normally distributed with mean $\mu = 21$ and $\sigma = 5$. To find the probability a randomly selected student scores above 27:

• Compute z-score: $z = \frac{27 - 21}{5} = 1.2$

• Find $P(X > 27) = P(Z > 1.2) = 1 - P(Z < 1.2) = 1 - 0.8849 = 0.1151$

Sample Means and the Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution, provided the population has finite mean and variance.

• Mean of sample means: $\mu_{\bar{x}} = \mu$

• Standard deviation (standard error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Example: For $\mu = 21$, $\sigma = 5$, $n = 36$, $\sigma_{\bar{x}} = \frac{5}{6} = 0.833$

Confidence Intervals

Confidence Interval for a Population Mean (Known $\sigma$)

A confidence interval estimates the range in which a population parameter lies, with a certain level of confidence (e.g., 95%).

• Formula: $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• $z^*$ is the critical value from the standard normal table for the desired confidence level (e.g., $z^* = 1.96$ for 95%).

Example: For $\bar{x} = 8.1$, $\sigma = 2.7$, $n = 100$, 95% CI: $8.1 \pm 1.96 \times \frac{2.7}{10} = 8.1 \pm 0.53 = (7.57, 8.63)$

Confidence Interval for a Population Mean (Unknown $\sigma$)

• Use the t-distribution when the population standard deviation is unknown and the sample size is small ($n < 30$).

• Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• $t^*$ is the critical value from the t-distribution with $n-1$ degrees of freedom.

Confidence Interval for a Population Proportion

• Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• $\hat{p}$ is the sample proportion.

Example: In a sample of 682, 218 rely on Social Security. $\hat{p} = \frac{218}{682} = 0.32$. For 95% CI: $0.32 \pm 1.96 \sqrt{\frac{0.32 \times 0.68}{682}} = 0.32 \pm 0.034 = (0.286, 0.354)$

Critical Values and Error

Finding Critical Values

• Critical values ($z^*$ or $t^*$) correspond to the desired confidence level.

• For a 95% confidence level, $z^* = 1.96$; for 90%, $z^* = 1.645$.

Margin of Error

• Formula for mean: $E = z^* \frac{\sigma}{\sqrt{n}}$

• Formula for proportion: $E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Summary Table: Confidence Interval Formulas

Parameter	Population SD Known	Population SD Unknown
Mean	$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$	$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$
Proportion	$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Additional info:

• When sample size is large ($n \geq 30$), the sample mean is approximately normally distributed even if the population is not normal (Central Limit Theorem).

• For small samples from non-normal populations, use nonparametric methods.