Sampling Distributions, Confidence Intervals, and One-Sample Hypothesis Testing

Sampling Distributions

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the distribution of sample proportions obtained from repeated random samples of a fixed size from a population. It is fundamental for making inferences about population proportions based on sample data.

• Sample Proportion (\( \hat{p} \)): The proportion of successes in a sample, calculated as \( \hat{p} = \frac{x}{n} \), where x is the number of successes and n is the sample size.

• Standard Error of the Sample Proportion: Measures the variability of the sample proportion from sample to sample. The formula is: $SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Normal Approximation: For large samples, the sampling distribution of \( \hat{p} \) is approximately normal if both \( np \geq 5 \) and \( n(1-p) \geq 5 \).

• Example: If 60 out of 200 surveyed customers prefer a new product, \( \hat{p} = 0.3 \). The standard error is calculated using the formula above.

Confidence Interval Estimation

Confidence Interval Estimate for the Population Mean (\( \sigma \) Known)

A confidence interval provides a range of values within which the population mean is likely to fall, with a specified level of confidence, when the population standard deviation is known.

• Level of Confidence: The probability that the interval estimate contains the true population parameter (commonly 90%, 95%, or 99%).

• Formula (Z-interval): $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ where \( \bar{x} \) is the sample mean, \( z_{\alpha/2} \) is the critical value from the standard normal distribution, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

• Example: For a sample mean of 50, \( \sigma = 10 \), \( n = 25 \), and 95% confidence, the interval is $50 \pm 1.96 \times 2 = (46.08, 53.92)$.

Confidence Interval Estimate for the Population Mean (\( \sigma \) Unknown)

When the population standard deviation is unknown, the t-distribution is used to construct the confidence interval for the mean.

• The t Distribution: A family of distributions that are wider than the normal distribution, used when estimating the mean from small samples.

• Degrees of Freedom (df): Calculated as \( n - 1 \), where \( n \) is the sample size.

• Formula (t-interval): $\bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}}$ where \( s \) is the sample standard deviation.

• Example: For \( \bar{x} = 100 \), \( s = 15 \), \( n = 9 \), and 95% confidence, use \( t_{0.025,8} \approx 2.306 \).

Confidence Interval Estimate for the Population Proportion

This interval estimates the true proportion of a population based on a sample proportion.

• Formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Example: If \( \hat{p} = 0.4 \), \( n = 100 \), and 95% confidence, the interval is $0.4 \pm 1.96 \times 0.049 = (0.304, 0.496)$.

Determining Sample Size

Calculating the required sample size ensures that estimates meet desired precision and confidence levels.

• Sample Size for the Mean: $n = \left( \frac{z_{\alpha/2} \sigma}{E} \right)^2$ where \( E \) is the desired margin of error.

• Sample Size for the Proportion: $n = \frac{z_{\alpha/2}^2 p(1-p)}{E^2}$

• Example: To estimate a mean with \( \sigma = 5 \), \( E = 1 \), and 95% confidence, $n = (1.96 \times 5 / 1)^2 = 96.04$, so at least 97 samples are needed.

Fundamentals of Hypothesis Testing: One-Sample Tests

Fundamentals of Hypothesis Testing

Hypothesis testing is a formal procedure for comparing observed data with a claim or hypothesis about a population parameter.

• Null Hypothesis (\( H_0 \)): The default assumption or claim to be tested (e.g., \( \mu = \mu_0 \)).

• Alternative Hypothesis (\( H_a \)): The competing claim (e.g., \( \mu \neq \mu_0 \), \( \mu > \mu_0 \), or \( \mu < \mu_0 \)).

• Type I Error (\( \alpha \)): Rejecting \( H_0 \) when it is true.

• Type II Error (\( \beta \)): Failing to reject \( H_0 \) when it is false.

• Level of Significance (\( \alpha \)): The probability of making a Type I error, commonly set at 0.05 or 0.01.

Z-Test for the Mean (\( \sigma \) Known)

• Test Statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• Decision Rule: Compare the calculated z to the critical value or use the p-value approach.

• Example: Testing if the mean is 100 with \( \sigma = 15 \), \( n = 36 \), and \( \bar{x} = 104 \).

Testing with Critical Values and p-Values

• Critical Value Approach: Reject \( H_0 \) if the test statistic falls in the rejection region defined by the critical value(s).

• p-Value Approach: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under \( H_0 \). Reject \( H_0 \) if p-value < \( \alpha \).

t-Test of Hypothesis for the Mean (\( \sigma \) Unknown)

• Test Statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Degrees of Freedom: \( n - 1 \).

• Example: For \( \bar{x} = 52 \), \( s = 8 \), \( n = 16 \), test if the mean differs from 50.

One-Tail Tests

• Left-Tailed Test: \( H_a: \mu < \mu_0 \)

• Right-Tailed Test: \( H_a: \mu > \mu_0 \)

• Decision Rule: Use the appropriate critical value for the direction of the test.

Z-Test of Hypothesis for the Proportion

• Test Statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Example: Testing if the proportion is 0.5 with \( \hat{p} = 0.56 \), \( n = 150 \).