Margin of Error & Sample Size for Confidence Interval | Statistics Tutorial #11| MarinStatsLectures

Summary

The video explains how to control the margin of error when constructing confidence intervals. Using an example of BMI measurements, the speaker demonstrates that the margin of error depends on three factors: the t-value, the sample standard deviation, and the sample size. The video focuses on strategies to reduce the margin of error, like increasing sample size, but acknowledges the trade-offs in real-world scenarios like time and cost. The speaker also discusses estimating standard deviation through literature or pilot studies to plan sample sizes more effectively.

Takeaways

• 📊 The margin of error (M.E.) in confidence intervals is determined by the t-value, sample standard deviation, and sample size.
• 🎯 To reduce the margin of error, you can either lower the t-value, but this decreases confidence, which may not be useful.
• ❌ Decreasing the standard deviation isn't feasible because it reflects the natural variability in the population, like BMI differences.
• 📈 Increasing the sample size is a practical way to decrease the margin of error, as it reduces variability in the estimate.
• 🧮 The formula for calculating sample size is: n = (t * s / M.E.)², where t is the t-value, s is the sample standard deviation, and M.E. is the desired margin of error.
• 🔍 For a 95% confidence level with a margin of error of 0.5, a sample size of 400 is required, using a t-value of 2 and a standard deviation of 5.
• 📅 Planning ahead is important: determine the margin of error you need before collecting data to avoid too-wide intervals.
• ⚖️ There's a balance between the ideal statistical world, where you collect enough data, and the real world, where time and resources are limited.
• 🔄 For large sample size calculations, it's common to replace the t-value with the z-value since they become similar in larger samples.
• 📚 If the standard deviation is unknown, you can estimate it by consulting similar studies, conducting a pilot study, or using expert knowledge and the 6-standard-deviation rule.

Q & A

What is the margin of error (M.E.) in the context of this example?

The margin of error (M.E.) is the amount by which the sample mean could vary from the true population mean. In this example, the M.E. is 2.5, which was calculated for a two-sided 95% confidence interval based on a sample mean of 25.2 and a sample standard deviation of 5.

Why might we want to decrease the margin of error?

We might want to decrease the margin of error to obtain a narrower, more precise confidence interval, providing a better estimate of the population mean with less uncertainty.

What are the three factors that control the margin of error?

The margin of error depends on three factors: the t-value (which relates to the confidence level), the sample standard deviation, and the sample size.

How can decreasing the t-value affect the margin of error?

Decreasing the t-value will reduce the margin of error, but this also decreases the confidence level. For example, using a 60% confidence level would result in a smaller M.E., but the interval would be less reliable, containing the true mean only 60% of the time.

Why is it not feasible to decrease the sample standard deviation to reduce the margin of error?

Decreasing the sample standard deviation is not feasible because it represents natural variability in the data. For example, in BMI measurements, people's BMIs naturally vary, and we cannot reduce this variability artificially.

How does increasing the sample size affect the margin of error?

Increasing the sample size reduces the margin of error because a larger sample provides a more accurate estimate of the population mean, making the confidence interval narrower.

What sample size is required to achieve a margin of error of 0.5 in this example?

To achieve a margin of error of 0.5 with a 95% confidence interval, a sample size of approximately 400 is required, assuming a sample standard deviation of 5 and a t-value of around 2.

Why is it important to plan for sample size ahead of time?

Planning the sample size ahead of time ensures that the margin of error meets the desired level of precision. If you collect too few samples, the confidence interval may be too wide to be useful, requiring more samples to improve the precision.

How can you estimate the sample standard deviation when planning for sample size?

To estimate the standard deviation ahead of time, you can either look at similar studies in the literature, conduct a small pilot study, or use expert knowledge to approximate the range of values and divide it by 6 if the data is normally distributed.

Why do we use the z-value instead of the t-value for larger sample size calculations?

When calculating sample sizes, we often use the z-value instead of the t-value because, for larger sample sizes, the t-distribution and z-distribution are approximately the same, simplifying the calculations without affecting the results.