Sampling Distributions, Confidence Intervals, and One-Sample Hypothesis Testing: Exam 3 Study Guide

Sampling Distributions

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the probability distribution of sample proportions for a given sample size, drawn from a population with a fixed proportion.

• Sample Proportion (\hat{p}): The ratio of the number of successes to the total sample size, 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: $SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ where p is the population proportion and n is the sample size.

• Example: If a population proportion is 0.4 and the sample size is 100, the standard error is: $SE_{\hat{p}} = \sqrt{\frac{0.4 \times 0.6}{100}} = \sqrt{0.0024} \approx 0.049$

Confidence Interval Estimation

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

A confidence interval provides a range of values within which the population mean is likely to fall, based on a sample mean and known population standard deviation.

• Level of Confidence: The probability that the interval estimate contains the population parameter. Common levels are 90%, 95%, and 99%.

• Using Z for the Confidence Interval: When \( \sigma \) is known, the confidence interval is: $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ 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 \( \bar{x} = 50 \), \( \sigma = 10 \), \( n = 25 \), and 95% confidence (\( z_{0.025} = 1.96 \)): $50 \pm 1.96 \frac{10}{5} = 50 \pm 3.92$ Interval: (46.08, 53.92)

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

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

• The t Distribution: A probability distribution used when estimating the mean of a normally distributed population with unknown standard deviation.

• Degrees of Freedom (df): For a sample, \( df = n - 1 \).

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

• Example: For \( \bar{x} = 50 \), \( s = 10 \), \( n = 25 \), and 95% confidence (\( t_{0.025,24} \approx 2.064 \)): $50 \pm 2.064 \frac{10}{5} = 50 \pm 4.128$ Interval: (45.872, 54.128)

Confidence Interval Estimate for the Population Proportion

Used to estimate the population proportion 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 (\( z_{0.025} = 1.96 \)): $0.4 \pm 1.96 \sqrt{\frac{0.4 \times 0.6}{100}} = 0.4 \pm 0.096$ Interval: (0.304, 0.496)

Determining Sample Size

Sample size determination ensures that the sample is large enough to achieve a desired margin of error for estimating the mean or proportion.

• 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 \( \sigma = 10 \), \( E = 2 \), and 95% confidence (\( z_{0.025} = 1.96 \)): $n = \left( \frac{1.96 \times 10}{2} \right)^2 = (9.8)^2 = 96.04$ So, at least 97 samples are needed.

Fundamentals of Hypothesis Testing: One-Sample Tests

Fundamentals of Hypothesis Testing

Hypothesis testing is a statistical method for making decisions about population parameters based on sample data.

• Null Hypothesis (H0): The statement being tested, usually a statement of no effect or no difference.

• Alternative Hypothesis (H1): The statement that contradicts the null hypothesis.

• Type I Error: Rejecting the null hypothesis when it is true (false positive).

• Type II Error: Failing to reject the null hypothesis when it is false (false negative).

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

• Z-statistic in Hypothesis Testing (\( \sigma \) is Known): $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$ where \( \mu_0 \) is the hypothesized population mean.

• Testing with Critical Value(s): Compare the test statistic to critical values to decide whether to reject H0.

• Testing with the p-value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under H0. If p-value < \( \alpha \), reject H0.

• Example: Testing if the mean is 50 with \( \bar{x} = 52 \), \( \sigma = 10 \), \( n = 25 \): $z = \frac{52 - 50}{10 / 5} = \frac{2}{2} = 1$ Compare to critical value for \( \alpha = 0.05 \) (two-tailed, \( z_{0.025} = 1.96 \)). Since 1 < 1.96, do not reject H0.

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

Used when the population standard deviation is unknown and the sample is drawn from a normally distributed population.

• t-statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where \( s \) is the sample standard deviation.

• Degrees of Freedom: \( df = n - 1 \).

• Example: For \( \bar{x} = 52 \), \( \mu_0 = 50 \), \( s = 10 \), \( n = 25 \): $t = \frac{52 - 50}{10 / 5} = 1$ Compare to critical value from t-table for \( df = 24 \).

One-tail Tests

One-tailed tests are used when the alternative hypothesis specifies a direction (greater than or less than).

• Right-tailed Test: H1: \( \mu > \mu_0 \)

• Left-tailed Test: H1: \( \mu < \mu_0 \)

• Critical region: All of \( \alpha \) is placed in one tail of the distribution.

• Example: If testing whether the mean is greater than 50, use a right-tailed test.

Z-Test of Hypothesis for the Proportion

Used to test hypotheses about population proportions.

• Z-statistic for Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ where \( \hat{p} \) is the sample proportion and \( p_0 \) is the hypothesized population proportion.

• Example: If \( \hat{p} = 0.4 \), \( p_0 = 0.5 \), \( n = 100 \): $z = \frac{0.4 - 0.5}{\sqrt{0.5 \times 0.5 / 100}} = \frac{-0.1}{0.05} = -2$ Compare to critical value for \( \alpha = 0.05 \) (two-tailed, \( z_{0.025} = 1.96 \)). Since -2 < -1.96, reject H0.