Key Definitions:
A population is the complete set of all individuals, items, or data of interest in a particular study. This is the entire group about which we want to draw conclusions.
A sample is a subset of the population that is actually observed or measured. We use samples that are randomly selected to make inferences about populations.
A parameter is a numerical characteristic that describes some aspect of a population (e.g., population proportion \(p\), population mean \(\mu\)).
A statistic is a numerical value calculated from sample data (e.g., sample proportion \(\hat{p}\), sample mean \(\bar{x}\)) used to estimate a population parameter.
Statistical inference is the process of constructing confidence intervals or testing hypotheses about a parameter.
Involves using sample statistics to: - Calculate point estimates (single best guess) - Construct confidence intervals (range of plausible values)
A level \(1-\alpha\) confidence interval for a population proportion is given by:
\[\hat{p} \pm z\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\] where \(\hat{p}\) is the sample proportion, \(n\) is the sample size, and \(z\) is the critical value dependent on \(\alpha\). If \(\alpha=0.05%\), \(z=1.96\).
Involves assessing evidence against a null hypothesis (\(H_0\)) using:
When testing a population proportion \(p\), the null hypothesis is always written as:
\[H_0: p = p_0\] and the alternative hypothesis can take one of the following three forms:
\[H_a: p < p_0\] \[H_a: p > p_0\]
or
\[H_a: p \neq p_0\] They are called the left-sided, right-sided, or two-sided alternative, respectively.
Note: Sometimes, \(H_a\) might be written as \(H_1\).
Source: https://wisc.pb.unizin.org/biocorestatistics/chapter/statistical-inference/
A confidence interval provides a range of plausible values for the true population proportion based on sample data. The general form is:
\[ \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Where: - \(\hat{p}\) = sample proportion - \(z\) = critical value from standard normal distribution. For the 95% confidence level, \(z\) is 1.96. For the 90% confidence level, \(z\) is 1.645. - \(n\) = sample size - The part \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) is called the standard error of the sample proportion \(\hat{p}\), usually denoted \(se\). - The part after \(pm\) is called the margin of error (\(m\)), which is the product of the critical value and the standard error.
Scenario: In a survey of 500 voters, 280 support Candidate A. Construct a 95% confidence interval for the true proportion of supporters.
prop.test(x=280, n=500, conf.level = 0.95, correct = FALSE)
1-sample proportions test without continuity correction
data: 280 out of 500, null probability 0.5
X-squared = 7.2, df = 1, p-value = 0.00729
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.5161969 0.6028882
sample estimates:
p
0.56 Interpretation: We are 95% confident that the true proportion of voters supporting Candidate A is between 51.62% and 60.28%.
Scenario: A factory tests 200 products and finds 12 defective. Construct a 90% CI for the defect rate.
prop.test(x=12, n=200, conf.level = 0.90, correct = FALSE)
1-sample proportions test without continuity correction
data: 12 out of 200, null probability 0.5
X-squared = 154.88, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
90 percent confidence interval:
0.03781444 0.09393107
sample estimates:
p
0.06 Interpretation: We are 90% confident the true defect proportion is between 3.78% and 9.39%.
Sample Size Requirements:
Margin of Error depends on:
Assumptions:
Hypothesis testing evaluates whether sample data provides sufficient evidence to reject a claim about a population proportion.
The general procedure:
State hypotheses:
Calculate test statistic:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
\]
Determine p-value based on alternative hypothesis. We will use R code to find the p-value, as the next section shows.
Compare p-value to the significance level \(\alpha\) (typically 0.05). If the p-value is no greater than the significance level, reject the null hypothesis. Otherwise, fail to reject it or don’t reject it.
Scenario: A company claims 30% of customers prefer their product. In a survey of 150 customers, 36 prefer it. Test if the true proportion differs from 30% (\(\alpha = 0.05\)).
prop.test(x=36, n=150, p = 0.30, alternative = "two.sided", correct = FALSE)
1-sample proportions test without continuity correction
data: 36 out of 150, null probability 0.3
X-squared = 2.5714, df = 1, p-value = 0.1088
alternative hypothesis: true p is not equal to 0.3
95 percent confidence interval:
0.1786931 0.3142914
sample estimates:
p
0.24 Results:
Scenario: A factory claims at most 5% of products are defective. In a batch of 300, 22 are defective. Test if the defect rate exceeds the claim (\(\alpha = 0.05\)).
prop.test(x=22, n=300, p = 0.05, alternative = "greater", correct = FALSE)
1-sample proportions test without continuity correction
data: 22 out of 300, null probability 0.05
X-squared = 3.4386, df = 1, p-value = 0.03184
alternative hypothesis: true p is greater than 0.05
95 percent confidence interval:
0.05220849 1.00000000
sample estimates:
p
0.07333333 Results:
Scenario: A manufacturer claims more than 95% defect-free parts. A sample of 50 parts gives 44 defect-free parts. Test if the defect-free rate is below the claim (\(\alpha = 0.05\)).
prop.test(x=46, n=50, p = 0.95, alternative = "less", correct = FALSE)
1-sample proportions test without continuity correction
data: 46 out of 50, null probability 0.95
X-squared = 0.94737, df = 1, p-value = 0.1652
alternative hypothesis: true p is less than 0.95
95 percent confidence interval:
0.000000 0.963578
sample estimates:
p
0.92 Results:
When estimating a 95% confidence interval for a population proportion, the maximum sample size \(n\) to achieve margin of error \(m\) is approximately \((\frac{1}{m})^2\).
Example: The maximum sample size \(n\) to achieve margin of error 0.12 at confidence level 95% is \((\frac{1}{0.12})^2=69.44\), or 70 (always rounded up).
A practical constraint limiting large sample size n is cost and time.
Definitions
Match each term to its correct definition:
| Term | Definition |
|---|---|
| Population | A) A numerical characteristic of a sample |
| Sample | B) The complete set of items of interest |
| Parameter | C) A subset of the population that is observed |
| Statistic | D) A numerical characteristic of a population |
True or False:
Calculation Practice
In a survey of 400 students, 160 reported using public transportation daily.
prop.test()# Your R code hereInterpretation
For the above CI (0.360, 0.440), explain what “90% confident” means in context.
Test Setup
A medication claims to be effective for 70% of patients. In a trial of 50 patients, 32 found it effective.
R Analysis
Conduct the test in R at α = 0.05 and interpret:
# Your test code hereCase Study
A website claims its conversion rate is 8%. In a sample of 200 visitor, 22 converted.
Error Analysis
If you reject H₀ when α = 0.05:
Sample Size Impact
Holding everything else constant, what happens to: