Small sample size confidence intervals | Probability and Statistics | Khan Academy

TL;DR

The video explains how to construct a 95% confidence interval to estimate the true expected blood pressure increase after taking a new drug.

Transcript

7 patients blood pressures have been measured after having been given a new drug for 3 months. They had blood pressure increases of, and they give us seven data points right here-- who knows, that's in some blood pressure units. Construct a 95% confidence interval for the true expected blood pressure increase for all patients in a population. So th... Read More

Key Insights

• 👶 Blood pressure increase after taking a new drug can be estimated with a confidence interval.
• ❓ The sample mean and sample standard deviation are used to estimate the population parameters.
• 🛩️ A t-distribution is utilized when dealing with small sample sizes.
• 🧡 The 95% confidence interval provides a range of values that likely contains the true population mean.
• ❓ Confidence intervals vary from sample to sample and are not fixed probabilities.
• 😃 The t-distribution accounts for uncertainty in estimating the standard deviation.
• 🍂 A 95% confidence interval means there is a 95% chance that the true mean falls within the interval.

Questions & Answers

Q: Why is it reasonable to assume a normal distribution for the population blood pressure increase?

Blood pressure increase is a biological process, and it is the result of many random events summed up, which tends to follow a normal distribution.

Q: How is the sample mean calculated?

The sample mean is obtained by summing up the blood pressure increases of the 7 patients and dividing by 7. In this case, the sample mean is 2.34.

Q: Why is the sample standard deviation used to estimate the true standard deviation of the population?

Since the true standard deviation is unknown, the sample standard deviation is a reasonable estimate based on the assumption that it is similar to the population standard deviation.

Q: How is the 95% confidence interval calculated using the t-distribution?

The t-distribution table is used to find the critical value for a 95% confidence level. The critical value is multiplied by the approximated standard deviation to determine the width of the confidence interval.

Summary & Key Takeaways

• The blood pressures of 7 patients were measured after taking a new drug for 3 months.

• The sample mean blood pressure increase was 2.34, and the sample standard deviation was 1.04.

• Due to the small sample size and unknown population distribution, a t-distribution is used to calculate the 95% confidence interval.