Statistics Test 3 Study Guide: Z-Scores, Confidence Intervals, Hypothesis Testing, and Proportions

Z-Scores and Normal Distribution

Standard Score (Z-Score)

The z-score is a statistical measure that describes a value's position relative to the mean of a distribution, measured in terms of standard deviations. It is commonly used to standardize scores and compare them across different distributions.

• Definition: The z-score indicates how many standard deviations a value is from the mean.

• Formula:

$z = \frac{X - \mu}{\sigma}$

• Interpretation: A positive z-score means the value is above the mean; a negative z-score means it is below the mean.

• Sample Mean Z-Score: For sample means, the z-score is calculated as:

$z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$

• Example: If the mean test score is 70, the standard deviation is 10, and a student scores 85, their z-score is $z = \frac{85 - 70}{10} = 1.5$.

Using the Z-Score Table

• Area to the Left: The z-table provides the probability that a value is less than a given z-score (area to the left).

• Area to the Right: To find the probability greater than a z-score, subtract the table value from 1.

• Two-Tailed Test: For two-tailed tests, sum the probabilities in both tails (extremes) of the distribution.

• Example: If the z-score is 1.96, the area to the left is approximately 0.975. The area to the right is $1 - 0.975 = 0.025$.

Proportions

Population and Sample Proportion

Proportions are used to describe the fraction of the population or sample with a particular characteristic.

• Population Proportion (p): The proportion of the entire population with a characteristic.

• Sample Proportion (\hat{p}): The proportion observed in a sample.

• Formulas:

$p = \frac{X}{N}$ $\hat{p} = \frac{x}{n}$

• Example: If 30 out of 100 surveyed students prefer online classes, the sample proportion is $\hat{p} = \frac{30}{100} = 0.3$.

Confidence Intervals

Margin of Error and Confidence Interval

A confidence interval estimates the range in which a population parameter lies, based on sample data. The margin of error quantifies the uncertainty in this estimate.

• Margin of Error for Means:

$E = z^* \frac{\sigma}{\sqrt{n}}$

• Margin of Error for Proportions:

$E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Constructing a 95% Confidence Interval:

For means: $\bar{X} \pm E$ For proportions: $\hat{p} \pm E$

• Example: If $\bar{X} = 50$, $\sigma = 10$, $n = 25$, and $z^* = 1.96$ for 95% confidence, then $E = 1.96 \times \frac{10}{5} = 3.92$. The interval is $50 \pm 3.92$.

Sample Size Calculation

To achieve a desired margin of error, calculate the required sample size.

• For Means:

$n = \left( \frac{z^* \sigma}{E} \right)^2$

• For Proportions:

$n = \frac{z^{*2} \hat{p}(1-\hat{p})}{E^2}$

• Example: To achieve $E = 0.05$ with $\hat{p} = 0.5$ and $z^* = 1.96$, $n = \frac{1.96^2 \times 0.5 \times 0.5}{0.05^2} = 384$.

Hypothesis Testing

Forming Hypotheses

Hypothesis testing involves making claims about population parameters and testing them with sample data.

• Null Hypothesis (H0): The default assumption (e.g., no effect, no difference).

• Alternative Hypothesis (Ha): The claim being tested (e.g., there is an effect).

• Example: H0: p = 0.5; Ha: p ≠ 0.5

Interpreting Results

• Rejecting H0: Evidence supports the alternative hypothesis.

• Failing to Reject H0: Not enough evidence to support the alternative hypothesis.

• Example: If p-value < 0.05, reject H0.

Significance Levels and P-Values

• Significance Level (\alpha): Common values are 0.05 and 0.01.

• P-Value: The probability of observing the sample result, or more extreme, if H0 is true.

• Decision Rule: If p-value < \alpha, the result is statistically significant.

• Example: If p-value = 0.03 and \alpha = 0.05, the result is significant.

Type I and Type II Errors

• Type I Error (\alpha): Incorrectly rejecting a true null hypothesis (false positive).

• Type II Error (\beta): Failing to reject a false null hypothesis (false negative).

• Example: Type I: Concluding a drug works when it does not. Type II: Concluding a drug does not work when it actually does.

Comparison Table: Type I vs. Type II Errors

Error Type	Definition	Symbol	Example
Type I	Rejecting a true null hypothesis	\alpha	False positive
Type II	Failing to reject a false null hypothesis	\beta	False negative

Summary Table: Key Formulas

Concept	Formula
Z-score	$z = \frac{X - \mu}{\sigma}$
Sample Mean Z-score	$z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$
Population Proportion	$p = \frac{X}{N}$
Sample Proportion	$\hat{p} = \frac{x}{n}$
Margin of Error (Mean)	$E = z^* \frac{\sigma}{\sqrt{n}}$
Margin of Error (Proportion)	$E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Sample Size (Mean)	$n = \left( \frac{z^* \sigma}{E} \right)^2$
Sample Size (Proportion)	$n = \frac{z^{*2} \hat{p}(1-\hat{p})}{E^2}$