### 8.1 A Single Population Mean Using the Normal Distribution

$\overline{X}~N\left({\mu}_{X},\frac{\sigma}{\sqrt{n}}\right)$ The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by

*EBM*, point estimate +

*EBM*)

*EBM* = $z\frac{\sigma}{\sqrt{n}}$ = the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.

*CL* = confidence level, or the proportion of confidence intervals created that is expected to contain the true population parameter

*α* = 1 – *CL* = the proportion of confidence intervals that will not contain the population parameter

${z}_{\frac{\alpha}{2}}$ = the *z*-score with the property that the area to the right of the *z*-score is $\frac{\propto}{2}$; this is the *z*-score, used in the calculation of *EBM*, where α = 1 – *CL*.

*n* = $\frac{{z}^{2}{\sigma}^{2}}{EB{M}^{2}}$ = the formula used to determine the sample size (*n*) needed to achieve a desired margin of error at a given level of confidence

General form of a confidence interval

(lower value, upper value) = (point estimate error bound, point estimate + error bound)

To find the error bound when you know the confidence interval,

error bound = upper value point estimate **or** error bound = $\frac{\text{uppervalue}-\text{lowervalue}}{2}\text{.}$

Single population mean, known standard deviation, normal distribution

Use the normal distribution for means; population standard deviation is known: *EBM* = *z*$\frac{\alpha}{2}\cdot \frac{\sigma}{\sqrt{n}}$.

The confidence interval has the format ($\overline{x}$ − *EBM*, $\overline{x}$ + *EBM*).

### 8.2 A Single Population Mean Using the Student's *t*-Distribution

*s* = the standard deviation of sample values

$t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ is the formula for the *t*-score, which measures how far away a measure is from the population mean in the Student’s *t*-distribution.

*df* = *n *– 1; the degrees of freedom for a Student’s *t*-distribution, where *n* represents the size of the sample

*T*~*t _{df}* the random variable,

*T*, has a Student’s

*t*-distribution with

*df*degrees of freedom

$EBM={t}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}$ = the error bound for the population mean when the population standard deviation is unknown

${t}_{\frac{\alpha}{2}}$ is the *t*-score in the Student’s *t*-distribution with area to the right equal to $\frac{\alpha}{2}\text{.}$

The general form for a confidence interval for a single mean, population standard deviation unknown, Student's *t* is given by

(lower bound, upper bound) = (point estimate – *EBM*, point estimate + *EBM*)

### 8.3 A Population Proportion

*p′ = x / n*, where *x* represents the number of successes and *n* represents the sample size. The variable *p*′ is the sample proportion and serves as the point estimate for the true population proportion.

${p}^{\prime}~N\left(p,\sqrt{\frac{pq}{n}}\right)$ The variable *p′* has a **binomial distribution** that can be approximated with the normal distribution shown here,

**Confidence interval for a proportion:**

$(\text{lowerbound,upperbound)}=({p}^{\prime}\u2013EBP,{p}^{\prime}+EBP)=\left({p}^{\prime}\u2013z\sqrt{\frac{{p}^{\prime}{q}^{\prime}}{n}},{p}^{\prime}+z\sqrt{\frac{{p}^{\prime}{q}^{\prime}}{n}}\right)$

$n=\frac{{z}_{\frac{\alpha}{2}}{}^{2}{p}^{\prime}{q}^{\prime}}{EB{P}^{2}}$ provides the number of participants needed to estimate the population proportion with confidence 1 – *α* and margin of error *EBP*.

Use the normal distribution for a single population proportion $p\prime =\frac{x}{n}\text{.}$

$EBP=\left({z}_{\frac{\alpha}{2}}\right)\sqrt{\frac{p\prime q\prime}{n}}p\prime +q\prime =1$

The confidence interval has the format (*p′* – *EBP*, *p′* + *EBP*).

$\overline{x}$ is a point estimate for *μ*.

*p′* is a point estimate for *ρ*.

*s* is a point estimate for *σ*.