Real-World Examples of Confidence Intervals in Business and Science.

📚 Quick Study Guide

🔬 Definition: A confidence interval estimates a population parameter based on a sample. It provides a range of values within which the true parameter is likely to fall. 📊 Formula (for population mean when population standard deviation is known): $ \bar{x} \pm z*(\frac{\sigma}{\sqrt{n}}) $, where:
• 📈 $ \bar{x} $ is the sample mean,
• 📉 $ z $ is the z-score corresponding to the desired confidence level,
• 🧪 $ \sigma $ is the population standard deviation, and
• 🔢 $ n $ is the sample size.
⚖️ Confidence Level: Represents the percentage of times that the true population parameter falls within the calculated interval. Common confidence levels are 90%, 95%, and 99%. 💡 Interpretation: A 95% confidence interval means that if we were to repeat the sampling process many times, 95% of the resulting intervals would contain the true population parameter. 🔑 Margin of Error: The range added and subtracted from the sample mean to create the interval, i.e., $ z*(\frac{\sigma}{\sqrt{n}}) $.

🧪 Practice Quiz

1. What does a 95% confidence interval mean?
   1. It means there is a 95% chance that the sample mean is equal to the population mean.
   2. It means that 95% of the data falls within the interval.
   3. It means that if we took many samples, 95% of the calculated intervals would contain the true population mean.
   4. It means there is a 5% chance that the population mean is outside the interval.
2. In market research, a company wants to estimate the average spending of customers. They survey 100 customers and find a sample mean of $50. If the population standard deviation is $10 and they want a 95% confidence interval, what is the margin of error? (z-score for 95% = 1.96)
   1. $0.98
   2. $1.96
   3. $2.50
   4. $9.80
3. A pharmaceutical company is testing a new drug. They find that, on average, the drug reduces symptoms by 20 points with a standard deviation of 5 points in a sample of 25 patients. Calculate the margin of error for a 99% confidence interval. (z-score for 99% = 2.576).
   1. 0.5152
   2. 1.288
   3. 2.576
   4. 2.88
4. Which of the following will result in a wider confidence interval?
   1. A larger sample size
   2. A smaller standard deviation
   3. A higher confidence level
   4. A lower confidence level
5. A political pollster surveys 500 voters and finds that 55% support a particular candidate. They want to create a 90% confidence interval for the true proportion of voters who support the candidate. What is the primary goal of creating this confidence interval?
   1. To determine the exact number of voters who support the candidate.
   2. To estimate the range within which the true proportion of supporters likely falls.
   3. To manipulate the public opinion about the candidate.
   4. To predict the election outcome with 100% accuracy.
6. A quality control engineer measures the lifespan of 40 light bulbs and calculates the mean lifespan to be 750 hours with a standard deviation of 50 hours. What can be done to reduce the width of the confidence interval without sacrificing confidence level?
   1. Increase the confidence level.
   2. Decrease the sample size.
   3. Increase the sample size.
   4. Use a biased sampling method.
7. In agriculture, a farmer wants to estimate the average yield of corn per acre. They randomly sample 16 acres and find an average yield of 180 bushels with a sample standard deviation of 20 bushels. Why is it important to use a confidence interval instead of just relying on the sample mean?
   1. The sample mean is always an accurate representation of the population mean.
   2. The confidence interval provides a range of plausible values, acknowledging sampling variability.
   3. Using a confidence interval guarantees a higher yield in the next harvest.
   4. Confidence intervals are only used for very large populations.