Lecture Notes on Confidence Intervals and Sample Size

Questions on Sections 7.1, 7.2, and 7.3

• The instructor opens the session by asking if there are any questions on sections 7.1, 7.2, and 7.3.

Determining the Type of Problem

• When approaching a problem, it is important to determine whether the data is numeric or categorical.

  • If the data is numeric, you will typically be dealing with means.

  • If the data is categorical, you will be dealing with proportions.

Numeric Data (Means)

• When working with means, determine whether the population standard deviation is known or if only the sample standard deviation is available.

  • If the population standard deviation is known, use a z-interval.

  • If only the sample standard deviation is known, use a t-interval.

Example Problem: T-Interval

• The example is from section 7.3, focusing on t-intervals. The problem involves finding a confidence interval for the mean.

• Given:

  • Sample size: 29

  • Sample mean: 3784

  • Sample standard deviation: 476

• Since only the sample standard deviation is provided, a t-interval is appropriate.

Steps to Solve

1. Go to stat then tests and choose option 8, TInterval.

2. Select the option to input summary statistics.

3. Enter the provided statistics:

   • Sample mean ($\bar{x}$): 3784

   • Standard deviation ($s$): 476

   • Sample size ($n$): 29

   • Confidence level ($C$): 99% (0.99)

4. Calculate to find the confidence interval.

Result

• The confidence interval is between $3540 and $4028 (rounded to the nearest dollar).

Interpretive Statement

• "We are 99% confident that the mean annual medical cost to treat insomnia is between $3540 and $4028."

Section 7.4: Proportions

• Section 7.4 covers proportions, where the calculator function used is one prop z interval.

• Proportions are used when dealing with categorical data rather than numeric data.

Example Scenario: Political Candidacy

• The instructor uses a hypothetical scenario of running for the made-up office of "dog catcher" in Norwalk.

• The question posed is whether voters support candidate Lalani (yes or no).

• In a survey of 300 voters, 93 support candidate Lalani. The sample proportion is 93/300 = 0.31 (31%).

Types of Proportion Problems

1. Constructing a confidence interval.

2. Finding the minimum sample size.

Constructing a Confidence Interval

• To construct a confidence interval, use the calculator function one prop z interval.

Steps to Solve

1. Go to stat, tests, and choose option A, one prop z interval.

2. The input screen will ask for x, n, and C-level.

   • x: Number of successes (93)

   • n: Sample size (300)

   • C-level: Confidence level (95% or 0.95)

Result

• The confidence interval is between 0.258 and 0.362.

Interpretation

• "We are 95% confident that the proportion of Norwalk voters who support Candidate Lalani is between 25.8% and 36.2%."

Calculating Proportions

• If not directly given, calculate the sample proportion (p) by dividing the number of successes by the total number of responses: $p = \frac{x}{n}$.

• $q = 1 - p$, where q represents the proportion of failures.

Example Problem

• 50% of 50 employers think employees are fairly paid, while only 21% of 200 employees agree. Find the 99% confidence interval for the proportion for each group.

Solution

1. Employers

   • x = 50% of 50 = 25

   • n = 50

2. Employees

   • x = 21% of 200 = 42

   • n = 200

Calculations and Interpretations

1. Employers:

   • Using one prop z interval with x = 25, n = 50, and C-level = 0.99 gives a confidence interval of 0.318 to 0.682.

   • Interpretation: "We are 99% confident that the proportion of employers who think people are fairly paid is between 31.8% and 68.2%."

2. Employees:

   • Using one prop z interval with x = 42, n = 200, and C-level = 0.99 gives a confidence interval of 0.136 to 0.284.

Minimum Sample Size

• When determining the minimum sample size, you are deciding how many individuals (people or things) to sample.

Formula

• The formula for minimum sample size (n) is derived from the margin of error portion of the confidence interval formula: $n = p \cdot q \cdot (\frac{z}{E})^2$

  • p: Sample proportion

  • q: 1 - p

  • z: Z-score (critical value)

  • E: Margin of error

Prior Estimate Available

• If a prior estimate of the proportion is available, use that value in the formula.

No Prior Estimate Available

• If no prior estimate is available, use $p = 0.5$ and $q = 0.5$.

Example Problem

• A recent study indicates that 23% of 194 women surveyed over the age of 55 were widows. Determine the minimum sample size needed for a 90% confidence interval with a margin of error within 5%.

Solution with Prior Estimate

1. Find the critical z-value for a 90% confidence interval: z = 1.645.

2. Identify the given values:

   • E = 0.05 (5% margin of error)

   • $p = 0.23$ (prior estimate)

   • $q = 0.77$

3. Plug the values into the formula:
   $n = 0.23 \cdot 0.77 \cdot (\frac{1.645}{0.05})^2$

4. Calculate:
   $n = 191.69$

5. Always round up. Therefore, $n = 192$.

Solution without Prior Estimate

1. Use $p = 0.5$ and $q = 0.5$.

2. Plug the values into the formula:
   $n = 0.5 \cdot 0.5 \cdot (\frac{1.645}{0.05})^2$

3. Calculate:
   $n = 270.6$

4. Round up. Therefore, $n = 271$.

Additional Notes

• Always round up in minimum sample size calculations.

• Ensure that percentages are converted to decimal form before using them in formulas.