Estimating Parameters and Determining Sample Sizes

Confidence Intervals for Population Proportions and Means

Confidence intervals provide a range of plausible values for a population parameter (such as a mean or proportion) based on sample data. The width of the interval depends on the sample size, variability, and confidence level.

• Confidence Interval (CI): An interval estimate, calculated from the sample, that is likely to contain the true population parameter with a specified probability (confidence level).

• Formula for CI for a Population Proportion:

• Formula for CI for a Population Mean (σ known):

• Formula for CI for a Population Mean (σ unknown):

• Interpretation: A 95% confidence interval means that if we were to take many samples and build a CI from each, about 95% of those intervals would contain the true parameter.

• Example: If a sample of 40 employees yields a sample proportion of 0.9, and the 95% CI is (0.795, 0.924), we are 95% confident that the true proportion of all employees who support the company is between 0.795 and 0.924.

Determining Sample Size for Estimation

To achieve a desired margin of error in a confidence interval, it is important to determine the minimum sample size required.

• Sample Size for Proportion:

• Sample Size for Mean:

• Where: is the desired margin of error, is the critical value for the confidence level, is the estimated proportion, and is the population standard deviation.

• Example: For a 99% confidence level, margin of error 0.04, and , the required sample size is approximately 529.

Hypothesis Testing

Steps in Hypothesis Testing

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

1. State the Null Hypothesis () and Alternative Hypothesis (): is the default claim; is what you seek evidence for.

2. Select the Significance Level (): Common choices are 0.05 or 0.01.

3. Determine the Test Statistic: Use , , or depending on the test and data.

4. Find the Critical Value(s) and Rejection Region(s): Based on and the test type (one- or two-tailed).

5. Compute the Test Statistic from Sample Data.

6. Make a Decision: If the test statistic falls in the rejection region, reject ; otherwise, do not reject .

7. State the Conclusion in Context.

Types of Hypothesis Tests

• Test for a Population Mean (σ known): Use -test.

• Test for a Population Mean (σ unknown): Use -test.

• Test for a Population Proportion: Use -test for proportions.

Key Terms and Concepts

• Test Statistic: A standardized value calculated from sample data, used to decide whether to reject .

• P-value: The probability, assuming is true, of obtaining a result at least as extreme as the observed one. If , reject .

• Critical Value: The threshold value that the test statistic must exceed to reject .

• Rejection Region: The set of values for the test statistic that leads to rejection of .

Example: Hypothesis Test for a Mean (σ unknown)

• Scenario: An oil change shop claims oil changes take under 15 minutes. A sample of 10 customers yields times: 7, 8, 12, 13, 14, 15, 15, 17, 18, 19 minutes.

• Hypotheses: ,

• Test Statistic:

• Decision: If is not in the rejection region, do not reject .

• Conclusion: There is not enough evidence to support the claim that the mean oil change time is under 15 minutes.

Summary Table: Hypothesis Test Components

Additional info:

• Some problems involve drawing and interpreting rejection regions and critical values on normal distribution curves.

• When population standard deviation is unknown and sample size is small, use the -distribution.

• Always check assumptions: normality, random sampling, and known/unknown population parameters.