17 pendugaan interval rata rata sampel kecil

Summary

This video discusses the estimation of confidence intervals for small sample means, specifically when the sample size is less than 30. The presenter explains the use of formulas for both unlimited and limited population samples, detailing calculations for standard deviation and sample averages. Through a practical example, the video demonstrates how to calculate the average time taken by employees in a company to complete a task, providing a 99% confidence interval for the results. The tutorial includes steps for using t-distribution tables and applying the formulas to arrive at the final estimated range for the mean.

Takeaways

• 😀 Small sample sizes (n < 30) require specific formulas for interval estimation of the mean.
• 😀 The formula for confidence interval estimation is: X̄ ± t(α/2) * (S / √n), where X̄ is the sample mean, t(α/2) is the t-statistic, and S is the sample standard deviation.
• 😀 The t-statistic is based on the desired confidence level and degrees of freedom (df = n - 1).
• 😀 The sample standard deviation (S) is crucial for calculating the margin of error in the confidence interval.
• 😀 For finite populations, the confidence interval formula is adjusted with a correction factor to account for sampling without replacement.
• 😀 The formula for finite populations is: X̄ ± t(α/2) * (S / √n) * √((N - n) / (N - 1)), where N is the population size.
• 😀 The degrees of freedom (df) is the sample size minus one (n - 1), which is used to determine the appropriate t-statistic value.
• 😀 A confidence level of 99% corresponds to a t-statistic value of approximately 3.35 for a sample size of 9 and 8 degrees of freedom.
• 😀 In the example given, the work times of 9 employees were used to calculate the sample mean and standard deviation.
• 😀 The final confidence interval for the average work time was between 13.99 and 18.24 minutes, indicating the range within which the true population mean is likely to fall.

Q & A

What is the main topic of the video?

The main topic of the video is about interval estimation for the mean, specifically for small sample sizes (n < 30), using the T-distribution to calculate confidence intervals.

What is the significance of using T-distribution in this context?

T-distribution is used because the sample size is small (less than 30), and the population standard deviation is unknown, making it more suitable than the normal distribution for calculating confidence intervals.

How do you calculate the sample mean (X̄)?

The sample mean (X̄) is calculated by summing all the individual sample values and then dividing by the number of samples (n).

What formula is used to calculate the standard error (SE)?

The standard error (SE) is calculated by dividing the standard deviation (S) by the square root of the sample size (n), i.e., SE = S / √n.

What is the formula used for calculating the confidence interval for the mean in the case of a small sample size?

The formula used is: X̄ ± t(α/2) * (S / √n), where X̄ is the sample mean, t(α/2) is the critical value from the T-distribution, S is the sample standard deviation, and n is the sample size.

What does the 99% confidence level imply in this context?

A 99% confidence level means that we are 99% confident that the true population mean lies within the calculated interval.

How do you find the critical t-value from the T-distribution table?

The critical t-value is found by determining the degrees of freedom (n - 1), looking up the corresponding value for the 99% confidence level (α = 0.01), and using the T-distribution table to get the appropriate t-value.

What is the value of the critical t-value in this example?

In this example, the critical t-value for a 99% confidence level and 8 degrees of freedom (n-1 = 9-1 = 8) is 3.35.

What is the importance of calculating the sample's sum of squares (ΣX²)?

The sum of squares (ΣX²) is used to calculate the variance and standard deviation of the sample, which are essential for calculating the standard error and ultimately the confidence interval.

How is the final confidence interval for the average time calculated in this example?

The final confidence interval is calculated using the formula: X̄ ± t(α/2) * (S / √n). In this example, the resulting interval is between 13.99 minutes and 18.24 minutes.