Solved Exercises on Inferences from Samples in Grade 10

Master statistical inference: confidence intervals, hypothesis testing, sampling distributions, and parameter estimation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Confidence Interval for Mean
Exercise 1
A sample of 40 students had an average GPA of 3.2 with a standard deviation of 0.4. Construct a 95% confidence interval for the population mean GPA.
Definition:

Confidence Interval: Range of values likely to contain the population parameter with specified confidence level

Formula: CI = x̄ ± z*(σ/√n) for large samples or x̄ ± t*(s/√n) for small samples

Margin of Error: Critical value × Standard error

Confidence interval method:
  1. Identify sample statistics (x̄, s, n)
  2. Determine confidence level and critical value
  3. Calculate standard error
  4. Construct interval: x̄ ± critical value × standard error
Given
n=40, x̄=3.2, s=0.4, CL=95%
Critical value
z=1.96
Interval
3.2 ± 0.124
Step 1: Identify sample statistics

Sample size (n) = 40, Sample mean (x̄) = 3.2, Sample standard deviation (s) = 0.4

Step 2: Determine critical value

For 95% confidence level, α = 0.05, α/2 = 0.025

Since n ≥ 30, use z-distribution: z₀.₀₂₅ = 1.96

Step 3: Calculate standard error

SE = s/√n = 0.4/√40 = 0.4/6.325 = 0.0632

Step 4: Calculate margin of error

ME = z × SE = 1.96 × 0.0632 = 0.124

Step 5: Construct confidence interval

CI = x̄ ± ME = 3.2 ± 0.124 = (3.076, 3.324)

95% CI: (3.076, 3.324)
Final answer:

We are 95% confident that the true population mean GPA is between 3.076 and 3.324

Applied rules:

• Large sample: Use z-distribution when n ≥ 30

• Confidence level: 95% means we expect 95% of intervals to contain true mean

• Standard error: SE = s/√n for sample standard deviation

2 Hypothesis Testing
Exercise 2
A manufacturer claims their light bulbs last 1000 hours on average. A sample of 25 bulbs had a mean life of 980 hours with a standard deviation of 50 hours. Test the claim at α = 0.05 significance level.
Definition:

Hypothesis Testing: Statistical procedure to evaluate claims about population parameters

Null Hypothesis (H₀): Statement of no effect or no difference (usually includes equality)

Alternative Hypothesis (H₁): Research hypothesis (opposite of null)

Given
n=25, x̄=980, s=50, μ₀=1000, α=0.05
Test statistic
t = -2.0
Decision
Fail to reject H₀
Step 1: State hypotheses

H₀: μ = 1000 (manufacturer's claim)

H₁: μ ≠ 1000 (two-tailed test)

Step 2: Select significance level

α = 0.05

Step 3: Calculate test statistic

Since n < 30, use t-distribution: t = (x̄ - μ₀)/(s/√n)

t = (980 - 1000)/(50/√25) = -20/(50/5) = -20/10 = -2.0

Step 4: Find critical value

Degrees of freedom = n - 1 = 24

For α = 0.05 (two-tailed), t₀.₀₂₅ = ±2.064

Step 5: Make decision

Since |t| = 2.0 < 2.064, we fail to reject H₀

Fail to reject H₀ at α = 0.05
Final answer:

At the 0.05 significance level, there is insufficient evidence to reject the manufacturer's claim that bulbs last 1000 hours on average

Applied rules:

• Small sample: Use t-distribution when n < 30

• Decision rule: Reject H₀ if |test statistic| > critical value

• Two-tailed test: Alternative is ≠, so divide α by 2

3 Sampling Distribution
Exercise 3
The average height of men in a population is 70 inches with a standard deviation of 3 inches. If samples of size 36 are taken, find the probability that the sample mean is between 69.5 and 70.5 inches.
Definition:

Sampling Distribution: Distribution of sample statistics (like sample mean)

Central Limit Theorem: Sample mean approaches normal distribution as n increases

Standard Error: Standard deviation of sampling distribution = σ/√n

Given
μ=70, σ=3, n=36, 69.5 < X̄ < 70.5
Std error
σ/√n = 0.5
Z-scores
Z₁=-1, Z₂=1
Step 1: Identify population parameters

Population mean (μ) = 70 inches, Population standard deviation (σ) = 3 inches

Step 2: Find sampling distribution parameters

Sample mean distribution: Mean = μ = 70 inches

Standard error = σ/√n = 3/√36 = 3/6 = 0.5 inches

Step 3: Convert bounds to z-scores

Lower bound: Z₁ = (69.5 - 70)/0.5 = -0.5/0.5 = -1

Upper bound: Z₂ = (70.5 - 70)/0.5 = 0.5/0.5 = 1

Step 4: Calculate probability

P(69.5 < X̄ < 70.5) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)

= 0.8413 - 0.1587 = 0.6826

P(69.5 < X̄ < 70.5) = 0.6826
Final answer:

The probability that the sample mean is between 69.5 and 70.5 inches is approximately 0.6826 or 68.26%

Applied rules:

• Sampling distribution: X̄ ~ N(μ, σ²/n)

• Standard error: σ_X̄ = σ/√n

• Central Limit Theorem: Applies for n ≥ 30

Statistical Inference Concepts
\(CI = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}}, t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, SE = \frac{\sigma}{\sqrt{n}}\)
Key Statistical Formulas
Confidence Interval
x̄ ± z·(s/√n)
Estimate population parameter
Test Statistic
t = (x̄ - μ₀)/(s/√n)
Test hypotheses
Standard Error
σ/√n
Spread of sampling distribution
Key definitions:

Population Parameter: True value for entire population (μ, σ, p)

Sample Statistic: Value calculated from sample (x̄, s, p̂)

Statistical Inference: Process of drawing conclusions about population based on sample

Inference methodology:
  1. Data collection: Obtain representative sample
  2. Parameter identification: Determine what to estimate/test
  3. Method selection: Choose appropriate statistical procedure
  4. Calculation: Compute test statistic or confidence interval
  5. Interpretation: Draw conclusion in context
Tip 1: Use z-distribution for large samples (n ≥ 30), t-distribution for small samples.
Tip 2: Larger sample sizes reduce standard error and increase precision.
Tip 3: Always interpret results in the context of the problem.
Tip 4: Confidence level indicates reliability, not probability of containing true value.
Common errors: Misinterpreting confidence level, wrong distribution choice, calculation mistakes.
Exam preparation: Practice identifying when to use z vs t, memorize formulas, work with real-world contexts.
Essential formulas:

• Confidence interval: x̄ ± z*(s/√n) or x̄ ± t*(s/√n)

• Test statistic: t = (x̄ - μ₀)/(s/√n)

• Standard error: SE = s/√n

• Degrees of freedom: df = n - 1

Solution: Exercises 4 to 5
4 Proportion Inference
Exercise 4
In a survey of 500 voters, 280 said they would vote for candidate A. Construct a 90% confidence interval for the proportion of voters who favor candidate A.
Definition:

Sample Proportion: p̂ = x/n where x is number of successes

Proportion Confidence Interval: p̂ ± z·√[p̂(1-p̂)/n]

Conditions: np̂ ≥ 5 and n(1-p̂) ≥ 5 for normal approximation

Given
n=500, x=280, CL=90%
Sample prop
p̂ = 0.56
CI
(0.523, 0.597)
Step 1: Calculate sample proportion

p̂ = x/n = 280/500 = 0.56

Step 2: Check conditions

np̂ = 500 × 0.56 = 280 ≥ 5 ✓

n(1-p̂) = 500 × 0.44 = 220 ≥ 5 ✓

Step 3: Find critical value

For 90% confidence, α = 0.10, α/2 = 0.05

z₀.₀₅ = 1.645

Step 4: Calculate standard error

SE = √[p̂(1-p̂)/n] = √[0.56 × 0.44/500] = √[0.2464/500] = √0.0004928 = 0.0222

Step 5: Calculate margin of error

ME = z × SE = 1.645 × 0.0222 = 0.0365

Step 6: Construct confidence interval

CI = p̂ ± ME = 0.56 ± 0.0365 = (0.5235, 0.5965)

90% CI: (0.523, 0.597)
Final answer:

We are 90% confident that the true proportion of voters who favor candidate A is between 52.3% and 59.7%

Applied rules:

• Proportion CI: p̂ ± z·√[p̂(1-p̂)/n]

• Normal approximation: Requires np̂ ≥ 5 and n(1-p̂) ≥ 5

• Standard error: For proportions, SE = √[p̂(1-p̂)/n]

5 Comparing Two Means
Exercise 5
Two teaching methods are tested. Method A (n₁=30, x̄₁=85, s₁=5) and Method B (n₂=25, x̄₂=82, s₂=6). Test if Method A is significantly better at α = 0.05.
Definition:

Two-Sample Test: Compare means of two independent populations

Test Statistic: t = (x̄₁ - x̄₂)/√[(s₁²/n₁) + (s₂²/n₂)]

Null Hypothesis: H₀: μ₁ - μ₂ = 0 (no difference)

Given
Group A: n₁=30, x̄₁=85, s₁=5; Group B: n₂=25, x̄₂=82, s₂=6
Test stat
t = 2.04
Decision
Reject H₀
Step 1: State hypotheses

H₀: μ₁ - μ₂ = 0 (no difference in means)

H₁: μ₁ - μ₂ > 0 (Method A is better)

Step 2: Calculate test statistic

t = (x̄₁ - x̄₂)/√[(s₁²/n₁) + (s₂²/n₂)]

t = (85 - 82)/√[(5²/30) + (6²/25)] = 3/√[0.833 + 1.44] = 3/√2.273 = 3/1.508 = 1.99

Step 3: Approximate degrees of freedom

For unequal variances, use Welch's formula or assume df ≈ smaller of (n₁-1, n₂-1) = 24

Step 4: Find critical value

For α = 0.05 (one-tailed), with df = 24: t₀.₀₅ ≈ 1.711

Step 5: Make decision

Since t = 1.99 > 1.711, we reject H₀

Reject H₀: Method A significantly better
Final answer:

At the 0.05 significance level, there is sufficient evidence to conclude that Method A produces significantly higher scores than Method B

Applied rules:

• Two-sample t-test: For comparing means of two independent groups

• One-tailed test: H₁: μ₁ > μ₂ (directional alternative)

• Unequal variances: Use pooled or unpooled method based on assumption

Detailed Summary: Inferences from Samples
\(t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}, ME = z \cdot SE, \text{ where } SE = \frac{s}{\sqrt{n}}\)
Two-Sample Test and Margin of Error
Comprehensive definitions:

Statistical Inference: The process of drawing conclusions about population parameters based on sample statistics

Point Estimate: Single value used to estimate a population parameter (e.g., x̄ estimates μ)

Interval Estimate: Range of values that likely contains the population parameter

Type I Error: Rejecting a true null hypothesis (false positive)

Type II Error: Failing to reject a false null hypothesis (false negative)

Complete inference methodology:
  1. Problem identification: Determine if estimating parameter or testing hypothesis
  2. Data analysis: Check assumptions and conditions
  3. Method selection: Choose appropriate statistical procedure
  4. Calculation: Compute test statistic or confidence interval
  5. Decision making: Compare to critical value or interpret interval
  6. Conclusion: State results in context of original problem
Tip 1: Always check conditions before performing inference procedures.
Tip 2: Larger sample sizes generally lead to more precise estimates.
Tip 3: Be careful with one-tailed vs two-tailed tests based on research question.
Tip 4: Interpret results in the context of the original problem, not just mathematically.
Common applications: Medical studies, market research, quality control, educational assessments.
Key considerations: Random sampling, independence, normality, sample size adequacy.
Essential formulas and rules:

• Single mean CI: x̄ ± t*(s/√n) or x̄ ± z*(σ/√n)

• Single mean test: t = (x̄ - μ₀)/(s/√n)

• Proportion CI: p̂ ± z·√[p̂(1-p̂)/n]

• Two-sample test: t = (x̄₁ - x̄₂)/√[(s₁²/n₁) + (s₂²/n₂)]

• Decision rule: Reject H₀ if |test stat| > critical value

• Confidence level: (1 - α) × 100%

Visualization: Statistical Inference Concepts
Inference Procedures Overview
Explore the different types of statistical inference procedures:
Confidence intervals, hypothesis tests, and sampling distributions
One-sample vs two-sample comparisons

Analysis: The chart shows different statistical inference procedures and their applications.

  • Confidence intervals provide ranges for population parameters
  • Hypothesis tests evaluate claims about population parameters
  • Sampling distributions describe variability of sample statistics

Questions & Answers

Question: What's the difference between a confidence interval and a hypothesis test? When should I use each?

Answer: The main differences are:

  • Confidence Interval: Provides a range of plausible values for a population parameter. Goal: Estimate the parameter
  • Hypothesis Test: Evaluates whether sample data supports a specific claim about a parameter. Goal: Make a decision about the parameter

Use confidence intervals when you want to estimate "what is the value?"

Use hypothesis tests when you want to determine "is there evidence for this claim?"

For example, if you want to know the average height of students, use a confidence interval. If you want to test if the average is different from 68 inches, use a hypothesis test.

Question: What does "fail to reject the null hypothesis" mean? Isn't that the same as accepting it?

Answer: "Fail to reject H₀" and "accept H₀" are NOT the same thing!

"Fail to reject H₀" means:

  • We don't have enough evidence to conclude that H₀ is false
  • The data is consistent with H₀
  • We cannot prove H₀ is true, only that we can't disprove it

Think of it like a legal trial: "not guilty" doesn't mean "innocent" - it means there wasn't enough evidence to prove guilt beyond reasonable doubt.

We never "accept" the null hypothesis; we only fail to reject it due to insufficient evidence.

Question: What's the difference between z-distribution and t-distribution? When do I use each?

Answer: The key differences are:

  • z-distribution: Known population standard deviation (σ), large samples (n ≥ 30)
  • t-distribution: Unknown population standard deviation (using sample s), small samples (n < 30)

The t-distribution has heavier tails than the z-distribution, accounting for additional uncertainty when using sample standard deviation.

As sample size increases, the t-distribution approaches the z-distribution.

Rule of thumb: Use t-distribution when σ is unknown (which is most real-world situations) and especially for small samples.

Question: How do I interpret a confidence interval correctly?

Answer: A common misconception is interpreting confidence intervals. Here's the correct way:

For a 95% confidence interval of (3.076, 3.324):

CORRECT interpretation: "We are 95% confident that the true population mean falls between 3.076 and 3.324."

INCORRECT interpretation: "There is a 95% probability that the true mean is between 3.076 and 3.324."

The confidence level refers to the reliability of the procedure over many samples, not the probability for a specific interval.

The true parameter is fixed; our interval either contains it or doesn't. The confidence level describes the long-run success rate of the method.

Question: What are Type I and Type II errors in hypothesis testing?

Answer: Type I and Type II errors occur when our conclusion differs from reality:

  • Type I Error: Rejecting a true null hypothesis (False Positive)
  • Probability: α (significance level)
  • Example: Convicting an innocent person
  • Type II Error: Failing to reject a false null hypothesis (False Negative)
  • Probability: β (depends on true parameter value)
  • Example: Acquitting a guilty person

In practice, we control α (Type I error rate) and try to minimize β by increasing sample size.

The power of a test is 1 - β, representing the probability of correctly rejecting a false null hypothesis.