Guided Practice
Work through statistics problems step-by-step with detailed explanations and scaffolded support.
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This guided practice lesson helps you apply concepts from probability distributions and hypothesis testing through carefully structured problems. Each problem includes hints, step-by-step solutions, and explanations of the reasoning process.
Key Strategies for Statistics Problems
- Identify the distribution type: Determine if you're working with normal, binomial, or other distributions
- List given information: Write down all known values, sample sizes, and parameters
- Choose the appropriate test: Match your hypothesis to the correct statistical test
- Check assumptions: Verify conditions for the test are met
- Calculate systematically: Follow the formula step-by-step
- Interpret in context: Explain what your answer means in real-world terms
Examples
Example 1: Normal Distribution Application
Problem: The heights of adult women in a population are normally distributed with mean 64.5 inches and standard deviation 2.5 inches. What proportion of women are between 62 and 67 inches tall?
Show Step-by-Step Solution
- Identify given values: mean = 64.5, sd = 2.5, x1 = 62, x2 = 67
- Calculate z-scores:
- z1 = (62 - 64.5) / 2.5 = -1.0
- z2 = (67 - 64.5) / 2.5 = 1.0
- Find probabilities: P(-1 < Z < 1) = 0.6827
- Answer: About 68.27% of women are between 62 and 67 inches tall.
Example 2: Hypothesis Test Setup
Problem: A company claims their batteries last at least 500 hours on average. A consumer group tests 36 batteries and finds a mean life of 490 hours with a standard deviation of 30 hours. Set up the hypothesis test at the 0.05 significance level.
Show Step-by-Step Solution
- State hypotheses:
- H0: mean >= 500 (claim is true)
- Ha: mean < 500 (claim is false)
- Identify test type: One-tailed z-test (n >= 30)
- Calculate test statistic: z = (490 - 500) / (30 / sqrt(36)) = -2.0
- Find critical value: z_critical = -1.645 (left-tailed, alpha = 0.05)
- Decision: Since -2.0 < -1.645, reject H0
- Conclusion: There is sufficient evidence to conclude the batteries last less than 500 hours on average.
Practice Problems
Work through these 10 problems. Try each one before revealing the solution.
Problem 1: Z-Score Calculation
Test scores are normally distributed with mean 75 and standard deviation 8. Find the z-score for a student who scored 91.
Show Answer
z = (91 - 75) / 8 = 16 / 8 = 2.0
This score is 2 standard deviations above the mean.
Problem 2: Probability from Z-Score
Using the standard normal distribution, find P(Z > 1.5).
Show Answer
P(Z > 1.5) = 1 - P(Z < 1.5) = 1 - 0.9332 = 0.0668 or about 6.68%
Problem 3: Confidence Interval
A sample of 100 students has a mean GPA of 3.2 with standard deviation 0.5. Construct a 95% confidence interval for the population mean.
Show Answer
CI = 3.2 +/- 1.96 * (0.5 / sqrt(100))
CI = 3.2 +/- 1.96 * 0.05 = 3.2 +/- 0.098
(3.102, 3.298)
Problem 4: Hypothesis Test Decision
For a two-tailed test with alpha = 0.05, the test statistic is z = 2.3. Should you reject the null hypothesis?
Show Answer
Critical values: +/- 1.96
Since |2.3| > 1.96, reject H0.
Problem 5: Sample Size Determination
You want to estimate a population proportion within 3% with 95% confidence. What minimum sample size is needed? (Use p = 0.5 for maximum variability)
Show Answer
n = (z^2 * p * (1-p)) / E^2
n = (1.96^2 * 0.5 * 0.5) / 0.03^2
n = 0.9604 / 0.0009 = 1068 (rounded up)
Problem 6: Standard Error
A population has standard deviation 20. What is the standard error of the mean for samples of size 64?
Show Answer
SE = sigma / sqrt(n) = 20 / sqrt(64) = 20 / 8 = 2.5
Problem 7: P-Value Interpretation
A hypothesis test yields a p-value of 0.03. At the 0.01 significance level, what is your conclusion?
Show Answer
Since p-value (0.03) > alpha (0.01), fail to reject H0.
Note: At alpha = 0.05, we would reject H0 since 0.03 < 0.05.
Problem 8: Normal Distribution Percentile
SAT scores are normally distributed with mean 1050 and standard deviation 200. What score corresponds to the 90th percentile?
Show Answer
z for 90th percentile = 1.28
x = mean + z * sd = 1050 + 1.28 * 200 = 1050 + 256 = 1306
Problem 9: Type I and II Errors
A medical test has alpha = 0.05. If the null hypothesis is that a patient is healthy, describe what a Type I error would mean in this context.
Show Answer
A Type I error occurs when we reject a true null hypothesis.
In context: Concluding a healthy patient has the disease (false positive). This happens 5% of the time when testing healthy patients.
Problem 10: Margin of Error
A poll of 400 voters shows 55% support a candidate. Calculate the margin of error at 95% confidence.
Show Answer
ME = z * sqrt(p(1-p)/n)
ME = 1.96 * sqrt(0.55 * 0.45 / 400)
ME = 1.96 * sqrt(0.000619) = 1.96 * 0.0249 = 0.0488 or about 4.9%
Check Your Understanding
Answer these questions to verify your mastery of the concepts:
- What is the relationship between confidence level and margin of error?
- When should you use a z-test versus a t-test?
- What does a p-value represent?
- How does sample size affect the standard error?
Show Key Answers
- Higher confidence level = wider interval = larger margin of error
- Use z-test when n >= 30 or population sd is known; use t-test when n < 30 and population sd is unknown
- P-value is the probability of obtaining results at least as extreme as observed, assuming H0 is true
- Larger sample size = smaller standard error (inverse square root relationship)
Next Steps
- Review any problems where you struggled
- Practice additional problems from the Word Problems lesson
- Note common errors to watch for in the Common Mistakes lesson
- When ready, take the Unit Quiz to assess your overall understanding