INFERENCE

Four-Step Process

1. STATE
2. PLAN
   1. Check conditions
3. DO
4. CONCLUDE

Unit 6 through 9 covers inference, which is the practice of analyzing statistics to infer properties of a population.

• Point estimates are single statistics based on sample data to estimate a parameter

Name	Parameter	Point Estimate
Variance	${\sigma }^2$	$s^2$
Standard Deviation	$\sigma$	$s$
Mean	$\mu$	$\bar{x}$
Size	$N$	$n$
Proportion (categorical)	$p$	$\hat{p}$

• The standard deviation of a statistic is called the standard error
• The $z$-score of a test statistic is calculated as:
  • $\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p(0))}{n}}}$
  • Standard deviation is calculated with $p$,

CONFIDENCE INTERVALS

• Confidence intervals are intervals expected to contain the true population parameter
  • $\alpha$ refers to the area outside of the confidence interval (critical level ($z^{\ast }/t^{\ast }$) is inv. CDF of $\alpha$)

$$CI=\hat{p}\pm [z^{\ast }\cdot S{E}_p]$$

• Margin of error refers to the portion inside the brackets $[z^{\ast }\cdot S{E}_p]$
  • Recall from Chapter 1 that $S{E}_p=\sqrt{\frac{pq}{n}}$

Written Answers

Respond to confidence interval questions in the following format:

We are [x]% confident that the true [parameter] of [context] is between [upper bound] and [lower bound].

Conditions for creating a confidence interval are that:

• the sample must be random
• the sample must be <10% of the population
• the sample must contain >10 p and q

Example – Create a Confidence Interval

A random sample of 200 taxi drivers found that 23% responded “YES” to a monthly income of greater than $5000.

We will create a confidence interval of 98%.

Conditions:

• The sample is random
• 200 is likely less than 10% of all taxi drivers
• More than 10 answered “YES” and “NO”

Creating the interval:

• $\alpha =0.02$ therefore the area at each tail = 0.01
• InvNormCD(area=0.99, σ=1, μ=0) >>> 2.326
• $z^{\ast }=2.326$
• $CI=0.23\pm [2.326\cdot \sqrt{\frac{(0.23)(0.77)}{200}}]$
• Bounds (0.161, 0.299)

Answering in context: “We are 98% confident that the true proportion of taxi drivers who have a monthly income greater than $5500 is between 16.1% and 29.9%.”

Example – Determine What Sample Size is Needed

The Delta School District wants to find the proportion of teachers who smoke by constructing a 95% CI with a margin of error of 2%.

We are determining sample size.

To find $z^{\ast }$:

• InvNormCD(area=0.975, σ=1, μ=0) >>> 1.960

Applying the formula for margin of error:

• $MOE=0.02=1.960\cdot \sqrt{\frac{pq}{n}}$

$\hat{p}(p\text{ and }q\text{ in example})$ can be substituted with 0.50 since it is unknown:

• $0.02=1.960\cdot \sqrt{\frac{(0.50)(0.50)}{n}}$

Solving for sample size $n$:

• $n=2401$

To make the MOE smaller:

• reduce the percent confidence to lower the confidence level
• obtain a sample with less variability
• obtain a larger sample

Null Hypothesis

• The hypothesis is always stated in terms of $p$ (not $\hat{p}$)
• Each test is conducted with the assumption of the null hypothesis ($H_0$)
  • ${\mu }_{\hat{p}}=p$
  • $x$ has no effect on $y$
  • ${\mu }_A={\mu }_B$
• The tested statement that is contrary to the null is the alternate hypothesis ($H_A$)
  • ${\mu }_A\ne {\mu }_b$
  • ${\mu }_A\le {\mu }_B$
  • ${\mu }_A\ge {\mu }_B$
• The $p$-value is the probability of the result occurring assuming the null is true
  • $H_0$ fails to be rejected if $p>\alpha$
  • $H_A$ is accepted if $p\le \alpha$

Errors

• Type I errors ($p=\alpha$) occur when $H_0$ is rejected but is actually true
• Type II errors ($p=\beta$) occur when $H_0$ fails to be rejected but is actually false
  • Power ($p=1-\beta$) is the probability that a test will reject the null when it is false
  • $\alpha$ and $\beta$ are inversely related

Increasing	P(Type I)	P(Type II)	Power
$\alpha$	$\uparrow$	$\downarrow$	$\uparrow$
$n$ (sample size)	$-$	$\downarrow$	$\uparrow$
${\mu }_0-{\mu }_a$ (observed difference)	$-$	$\downarrow$	$\uparrow$

Choosing Inference Procesures

Qualitative Data

• ${\chi }^2$ homogeneity: multiple samples
• ${\chi }^2$ independence: one sample; one variable
• ${\chi }^2$ goodness-of-fit: one sample; two variables

Quantitative Data

Proportions:

• 1 proportion $z$: one sample
• 2 proportion $z$: two samples

Means:

• 1 sample $t$: one sample
• 2 sample $t$: two samples
• Matched pair $t$: two grouped samples

Slopes:

• Linear regression $t$