\section*{Statistics Practice Problems: Probability & Data Analysis}

Practice essential statistical concepts with these comprehensive problems covering probability, hypothesis testing, and data analysis.

\subsection*{Probability Problems}

\subsubsection*{Problem 1: Basic Probability} A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a red marble?

\textbf{Solution:} Total marbles = 5 + 3 + 2 = 10 $P(\text{red}) = \frac{5}{10} = 0.5$ or 50%

\subsubsection*{Problem 2: Conditional Probability} In a class of 30 students, 18 are girls and 12 are boys. If 8 girls and 5 boys wear glasses, what is the probability that a randomly selected student wears glasses?

\textbf{Solution:} Total students with glasses = 8 + 5 = 13 $P(\text{glasses}) = \frac{13}{30} \approx 0.433$ or 43.3%

\subsection*{Descriptive Statistics}

\subsubsection*{Problem 3: Mean and Standard Deviation} Calculate the mean and standard deviation for the dataset: 12, 15, 18, 20, 22, 25

\textbf{Solution:} Mean = $\frac{12 + 15 + 18 + 20 + 22 + 25}{6} = \frac{112}{6} = 18.67$

Variance = $\frac{\sum (x - \mu)^2}{n}$ = $\frac{[(12-18.67)^2 + (15-18.67)^2 + (18-18.67)^2 + (20-18.67)^2 + (22-18.67)^2 + (25-18.67)^2]}{6}$ = 20.22

Standard Deviation = $\sqrt{20.22} \approx 4.50$

\subsection*{Hypothesis Testing}

\subsubsection*{Problem 4: Z-Test} A sample of 50 students has a mean score of 75 with a standard deviation of 10. Test the hypothesis that the population mean is 70 at $\alpha = 0.05$.

\textbf{Solution:} $H_0: \mu = 70$ $H_1: \mu \neq 70$

$Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{75 - 70}{10/\sqrt{50}} = \frac{5}{1.414} \approx 3.54$

Critical value for $\alpha = 0.05$ is $\pm 1.96$ Since $|3.54| > 1.96$, reject $H_0$

\subsection*{Confidence Intervals}

\subsubsection*{Problem 5: 95% Confidence Interval} A sample of 100 people has a mean height of 170 cm with a standard deviation of 10 cm. Find the 95% confidence interval for the population mean.

\textbf{Solution:} 95% CI = $\bar{x} \pm (z \times \sigma/\sqrt{n})$ = $170 \pm (1.96 \times 10/\sqrt{100})$ = $170 \pm 1.96$ = $(168.04, 171.96)$

\subsection*{Keywords} statistics practice, probability problems, hypothesis testing, confidence intervals

\subsection*{Last Updated} July 12, 2025, 05:38 PM +04