Statistics Review: Probability Distributions, Binomial & Normal Distributions, Proportions, Hypothesis Testing, and Chi-Square Tests

Probability Distributions

Theoretical and Frequentist Distributions

Probability distributions describe the relative frequencies of outcomes of a random event. They are essential models for random experiments in statistics.

• All possible outcomes of a random experiment must be listed.

• Probability of each outcome must be specified.

• Sum of all probabilities must equal 1 (unity).

Example: Probability distribution for rolling a fair die:

Number	1	2	3	4	5	6
Probability	1/6	1/6	1/6	1/6	1/6	1/6

Random Variables

Discrete Random Variables

Discrete random variables take on countable values.

• Examples: Number of siblings, number of contacts stored on a phone.

Continuous Random Variables

Continuous random variables can take any value within a range and are not countable.

• Examples: Height of basketball players, weight of an object, duration of phone calls.

Normal Distribution and Z-Scores

Finding Probabilities Using the Standard Normal Distribution

To find probabilities for normally distributed variables:

1. Convert values to z-scores using the formula: $z = \frac{x - \mu}{\sigma}$

2. Look up the corresponding area (probability) in the standard normal table (z-table).

Example: For $x = 625$ grams, $\mu = 650$, $\sigma = 20$: $z = \frac{625 - 650}{20} = -1.25$ Probability for $z < -1.25$ is 0.1056.

Using Technology (TI Calculators)

1. Compute z

2. Press 2ND VARS

3. Scroll to normalcdf(

4. Enter limits and parameters

5. Output is the p-value (probability)

Binomial Distribution & Discrete Random Variables

Conditions for Binomial Distribution

• Fixed number of trials ($n$)

• Each trial has only two possible outcomes (success/failure)

• Trials are independent

• Probability of success ($p$) is constant for each trial

Binomial Probability Formula

The probability of getting $x$ successes in $n$ independent trials:

Use binomial formula, binomial tables, or technology (e.g., binompdf(n, p, x) on calculators).

Notation: $b(n, p, x)$

Binomial Syntax: binompdf(n, p, x)	x	P(x)
For x=0; binompdf(4, 0.3, 0)	0	0.240
For x=1; binompdf(4, 0.3, 1)	1	0.412
For x=2; binompdf(4, 0.3, 2)	2	0.265
For x=3; binompdf(4, 0.3, 3)	3	0.076
For x=4; binompdf(4, 0.3, 4)	4	0.008

Sampling Distributions & Confidence Intervals: Proportion

Sample Proportions: Conditions

• Sampling is random and independent

• Sample size is large enough: $np \geq 10$ and $n(1-p) \geq 10$

• Population is at least 10 times larger than the sample (if sampling without replacement)

Sample Proportions: Conclusions

• Sampling distribution of $\hat{p}$ is approximately normal

• Mean: $p$

• Standard deviation: $SD = \sqrt{\frac{p(1-p)}{n}}$

• Estimated standard deviation: $SD_{est} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Hypothesis Testing for One Sample: Proportions

Steps for Hypothesis Testing

1. Hypothesize Formulation: State $H_0$ and $H_A$ (e.g., $H_0: p = 0.40$, $H_A: p > 0.40$)

2. Prepare - Check CLT Conditions: Choose significance level $\alpha$, check randomness, sample size, and population size

3. Compute (Statistics) to Compare: Compute $z = \frac{\hat{p} - p}{SD}$, then find p-value

4. Interpret and Conclude: Compare p-value with $\alpha$; if p-value $<$ $\alpha$, reject $H_0$

Example: If 40% of a sample of 200 own dogs, test if this is higher than 34% (p-value = 0.0367, so reject $H_0$ at $\alpha = 0.05$)

Type I and Type II Errors

• Type I Error: Rejecting $H_0$ when it is true

• Type II Error: Not rejecting $H_0$ when it is false

Confidence Intervals (CI) for Proportions

Constructing a Confidence Interval

1. Check CLT conditions (random, large sample, large population)

2. Calculate estimated standard deviation: $SD_{est} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

3. Find $z^*$ for the desired confidence level (e.g., $z^* = 1.645$ for 90%)

4. Margin of error: $m = z^* \times SD_{est}$

5. CI: $\hat{p} \pm m$

Example: For $n=800$, $\hat{p}=0.35$, 90% CI is (0.3222, 0.3778)

Sample Size for Desired Margin of Error

To achieve a margin of error $m$ at confidence level $z^*$:

$n = \frac{1}{4} \left( \frac{z^*}{m} \right)^2$

Sampling Distributions & Confidence Intervals: Mean

Population and Sample Variances

• Sample Variance: $\frac{\text{Population Variance}}{n}$

• Sample Standard Deviation: $\sqrt{\text{Sample Variance}}$

Sample Mean: Conditions

• Random sample

• Normality (population normal or $n$ large, usually $n \geq 25$)

• Population at least 10 times larger than sample

Hypothesis Testing for One Sample: Means

Tailed vs. Two-Tailed Tests

Two-Tailed	One-Tailed (Left)	One-Tailed (Right)
$H_0: \mu = \mu_0$ $H_A: \mu \neq \mu_0$	$H_0: \mu = \mu_0$ $H_A: \mu < \mu_0$	$H_0: \mu = \mu_0$ $H_A: \mu > \mu_0$

Steps for Hypothesis Testing (Means)

1. Formulate hypotheses ($H_0$, $H_A$)

2. Check CLT conditions (random, normality, large population)

3. Compute test statistic: $t = \frac{\bar{x} - \mu_0}{SD_{est}}$, where $SD_{est} = \frac{s}{\sqrt{n}}$

4. Find p-value and compare with $\alpha$

5. Interpret and conclude

Example: Testing if mean years of experience among nurses has increased (p-value = 0.0187, so reject $H_0$ at $\alpha = 0.05$)

Chi-Square Tests & Goodness of Fit

Chi-Square Statistic

Measures how much observed frequencies differ from expected frequencies.

Formula:

$\chi^2_{obs} = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$

Example Table: Observed vs. Expected

	Girls	Boys	Total
Gym (Observed)	20	50	70
No Gym (Observed)	30	200	230
Total	50	250	300

	Girls	Boys
Gym (Expected)	11.67	58.33
No Gym (Expected)	38.33	191.67

Expected Frequency: $\frac{\text{(Row Total)} \times \text{(Column Total)}}{\text{Grand Total}}$

Calculated $\chi^2_{obs} = 9.31$

Conditions for Chi-Square Tests

• Data must be counts for categories

• Random sample

• Expected cell frequency at least 5

• Counts are independent

Additional info:

• These notes cover core topics from chapters 6-12 of a typical introductory statistics course, including probability distributions, binomial and normal distributions, sampling distributions, confidence intervals, hypothesis testing, and chi-square tests.