A randomly selected 14 people were asked how long they slept at night. The mean time was 5.5 hours, and the standard deviation was 0.77 hour. Find the 80% confidence interval of the mean time. Assume the variable is normally distributed.

a. The $t_{\alpha/2}$ at 80% confidence level is equal to:
b. Find the best point estimate of the population mean.

Detailed Answer with Explanation:

Calculating an 80% Confidence Interval for Average Sleep Time

Sleep is a vital part of daily life, and understanding how much sleep people get on average is an important measure of overall well-being. In this blog, we’ll calculate the 80% confidence interval for the average sleep time, using data from a sample of 14 randomly selected individuals. This calculation allows us to estimate the population mean based on the sample data.

Problem Overview

We are given the following data:

• Sample Mean Sleep Time $ (\bar{x}) = 5.5 $ hours
• Sample Standard Deviation $ (s) = 0.77 $ hours
• Sample Size $ (n) = 14 $

Our goal is to calculate:

• (a) The $ t_{\frac{\alpha}{2}} $ value at the 80% confidence level.
• (b) The best point estimate of the population mean, which is the sample mean.

Step-by-Step Solution

1. a) Calculating the $ t_{\frac{\alpha}{2}} $ Value at the 80% Confidence Level

The $ t_{\frac{\alpha}{2}} $ value corresponds to the critical value from the t-distribution that will be used to construct the confidence interval.

1. Determine the Significance Level:The confidence level is 80%, so the significance level $ (\alpha) $ is:$ \alpha = 1 – 0.8 = 0.2 $
2. Calculate Degrees of Freedom:The degrees of freedom $ (df) $ are calculated as:$ df = n – 1 = 14 – 1 = 13 $
3. Find the Critical Value:The critical value $ (t_c) $ for $ \alpha = 0.2 $ with 13 degrees of freedom is:$ t_{\frac{\alpha}{2}, df} = t_{0.2/2, 13} \approx 1.35 $

This value can be found using a t-distribution table or using Excel’s function T.INV.2T(0.2, 13).

2. b) Confidence Interval Calculation

Next, we use the following formula to calculate the confidence interval:

$ CI = \bar{x} \pm t_c \times \frac{s}{\sqrt{n}} $

Where:

• $ \bar{x} $ is the sample mean,
• $ t_c $ is the critical value from the t-distribution,
• $ s $ is the sample standard deviation,
• $ n $ is the sample size.

Substituting the values we have:

$ CI = 5.5 \pm 1.35 \times \frac{0.77}{\sqrt{14}} $

First, calculate the standard error:

$ \frac{0.77}{\sqrt{14}} \approx 0.2057 $

Now, multiply by the critical value:

$ 1.35 \times 0.2057 \approx 0.2777 $

Finally, calculate the confidence interval:

$ CI = 5.5 \pm 0.2777 $

Thus, the confidence interval is approximately:

$ CI = (5.22, 5.78) $

This means we are 80% confident that the true population mean of the sleep time falls between 5.22 hours and 5.78 hours.

Best Point Estimate of the Population Mean

The best point estimate of the population mean is the sample mean itself, which is:

$ \bar{x} = 5.5 $ hours.

Conclusion

In this example, we calculated the 80% confidence interval for the average sleep time of 14 individuals. Based on the sample data, we can be 80% confident that the true population mean of sleep time lies between 5.22 hours and 5.78 hours. Understanding confidence intervals helps in estimating population parameters and provides insights into how representative our sample is of the overall population.