Mini-Textbook Study Notes: Hypothesis Testing with One Sample and Related Distributions

Hypothesis Testing with One Sample

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing observed data to what is expected under a specific hypothesis.

• Null Hypothesis (H0): The default assumption that there is no effect or no difference.

• Alternative Hypothesis (H1 or Ha): The statement we are trying to find evidence for.

• Test Statistic: A standardized value calculated from sample data, used to decide whether to reject H0.

• p-value: The probability of obtaining a result at least as extreme as the observed one, assuming H0 is true.

Example: Testing whether the mean score of a sample differs from a known population mean.

Sampling Distributions and the Central Limit Theorem

Sampling Distributions

A sampling distribution is the probability distribution of a given statistic based on a random sample. The most common is the sampling distribution of the sample mean.

• As the number of samples increases, the sampling distribution of the mean becomes more normal (Central Limit Theorem).

• The mean of the sampling distribution equals the population mean.

• The standard deviation of the sampling distribution is called the standard error.

Formula for Standard Error of the Mean:

$SE = \frac{\sigma}{\sqrt{n}}$

Where $\sigma$ is the population standard deviation and $n$ is the sample size.

Example: If $\sigma = 2$ and $n = 50$, then $SE = \frac{2}{\sqrt{50}} \approx 0.2828$.

Confidence Intervals

Confidence intervals provide a range of values within which the population parameter is expected to lie with a certain probability (confidence level).

• For a 95% confidence interval: $\mu \pm 1.96 \times SE$

Example: For $\mu = 10$, $SE = 0.2828$, the 95% confidence interval is $10 \pm 1.96 \times 0.2828 = [9.446, 10.554]$.

z-Test for a Single Sample Mean

When Population Standard Deviation is Known

The z-test is used when the population standard deviation ($\sigma$) is known. It compares the sample mean to the population mean to determine if they are significantly different.

z-Test Statistic Formula:

$z = \frac{\bar{x} - \mu}{SE} = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

• If $|z|$ is greater than the critical value (e.g., 1.96 for 95% confidence), we reject H0.

• p-values can be found using statistical tables or software (e.g., Excel's NORM.S.DIST function).

Example: Sample mean = 11.9, population mean = 10, $\sigma = 2$, $n = 50$.

$z = \frac{11.9 - 10}{2/\sqrt{50}} = \frac{1.9}{0.2828} \approx 6.72$

This z-value is much greater than 1.96, so the difference is significant.

z-Test for Proportions

Comparing Proportions

The z-test can also be used to compare sample proportions to a population proportion.

z-Test Statistic for Proportions:

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

• $\hat{p}$ = sample proportion

• $p$ = population proportion

• $n$ = sample size

Example: Population proportion = 0.37, sample proportion = 0.30, $n = 20$.

t-Test for a Single Sample Mean

When Population Standard Deviation is Unknown

When $\sigma$ is unknown, the t-test is used. The t-distribution is similar to the normal distribution but has heavier tails, which accounts for extra uncertainty from estimating $\sigma$ with the sample standard deviation ($s$).

• Degrees of Freedom (df): $df = n - 1$

• The critical value of t depends on df and the desired confidence level.

t-Test Statistic Formula:

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

Example: Sample mean = 12, population mean = 10, $s = 2$, $n = 10$.

$t = \frac{12 - 10}{2/\sqrt{10}} = \frac{2}{0.6325} \approx 3.16$

Compare this value to the critical t-value from the t-table for $df = 9$ (e.g., 2.262 for $\alpha = 0.05$, two-tailed).

Additional info: The t-table provides critical values for different degrees of freedom and significance levels. Use the appropriate value for your test.

Chi-Square Test for Variance

Testing Variance

The chi-square ($\chi^2$) test is used to compare the variance of a sample to a known or hypothesized population variance. It is sensitive to normality assumptions.

Chi-Square Test Statistic:

$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

• Used to test if the sample variance is significantly different from the population variance.

• The distribution is not symmetric and depends on the degrees of freedom ($df = n-1$).

Additional info: The shape of the chi-square distribution changes with degrees of freedom. For small df, it is highly skewed; as df increases, it becomes more symmetric.

Summary Table: Key Hypothesis Tests

Test	When to Use	Test Statistic	Distribution
z-test (mean)	Known $\sigma$	$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$	Normal
t-test (mean)	Unknown $\sigma$	$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$	t-distribution
z-test (proportion)	Proportions	$z = \frac{\hat{p} - p}{\sqrt{p(1-p)/n}}$	Normal
Chi-square test	Variance	$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$	Chi-square

Conclusion

Single-sample hypothesis tests are foundational tools in statistics, allowing researchers to make inferences about population parameters based on sample data. Understanding when and how to use z, t, and chi-square tests is essential for proper statistical analysis.