Confidence Intervals: One Sample Mean t


Introduction

One Sample Mean t confidence intervals are used when... 

  • You are dealing with a single sample mean (\(\bar{x}\))
  • The SAMPLE standard deviation (\(s\)) is known, but the population standard deviation is NOT

Because the population standard deviation is unknown, the critical value used will be t* (as opposed to z*).

* For more information on the t-distribution, click here.

Sometimes, the problems will state the sample mean and sample standard deviation. However, oftentimes these problems will provide a data table instead, and you will have to plug the numbers into your calculator to find the sample mean and sample standard deviation. 

 

Example

You want to know the mean total serum cholesterol level for adults. You randomly select 46 adults, test them, and record each level. After analyzing the data, you find that the sample mean is 191.57 and the sample standard deviation is 8.4. Construct a 90% cofidence interval for the mean cholesterol level in adults. 

 

Step 1: Name the Confidence Interval: One Sample Mean t

Step 2: Check the Conditions 

1. Data is drawn from a random sample. 

2. The sampling distribution of  is approximately normal.

3. N ≥ 10n

Step 3: Construct the Interval (Apply the Formula) 

1 Sample Mean t Confidence Interval Formula:  \(\bar{x}\) \(^+_-\) \((t^*)({s \over \sqrt {n}})\)

\(\bar{x}\)  \(^+_-\)  \((t^*)({s \over \sqrt {n}})\)

→  191.57  \(^+_-\) \((1.679)({8.4 \over \sqrt {46}})\)

→  191.57  \(^+_-\)  2.079

191.57 - 2.079  = 189.49

191.57 + 2.079  = 193.65

Interval: (189.49, 193.65)

 

Note: The critical value was found using a t-table. For t*, the critical value changes depending on the degrees of freedom. Remember:

For the t-distribution, degrees of freedom = \(n - 1\)

A portion of the t-table is listed below with the part needed for our problem highlighted:

df = 40 1.303 1.684 2.021
df = 45 1.301 1.679 2.014
df = 50 1.299 1.676 2.009
  80% 90% 95%

 

 

Step 4: State the Conclusion

Based on the data, I am 90% confident that the  mean total serum cholesterol level for adults is between 189.49 and 193.65.