**Confidence Interval for Mean, µ.**

**Large-sample 100*(1-α)% Confidence interval using Z-distribution**

where** Z_{α/2 }**is the z-value corresponding to an area α/2 in the tail of the standard normal distribution, and σ

_{x ̅}is the standard deviation of the sampling distribution, σ is the population standard deviation, and

*s*is the standard deviation of the sample.

Conditions required for a valid large-sample confidence interval for *µ*

- A random sample is selected from the target population
- The sample size
*n*is large (*n*> 30). Due to the Central Limit Theorem, this condition insures that the sampling distribution is approximately normal and that*s*will be a good estimator of*σ*.

**Commonly used values of Z_{α/2}**

**Small Samples**

With small samples, n < 30, use of the standard normal *z* as a test statistic is problematic for making an inference about the mean µ.

- The Central Limit Theorem only applies for large samples; therefore, we cannot assume the sampling distribution of x-bar is approximately normal.
- The population standard deviation, σ, is almost never known except for academic problems.

Instead of the standard normal statistic, use the t-statistic:

where *s* is the sample standard deviation.

The t-distribution has a mean of 0 like the z-distribution, but its shape depends on the sample size *n*. In the t-distribution tables, we use the degrees of freedom, *n* – 1, and α to find the value of the t-statistic, *t _{α}*.

The reason we set 30 as the determinant of whether to use the *z* or the *t* distribution, is that as *n *increases, the difference between the t-statistic and are essentially identical.

To form the confidence interval for a small sample from a normal distribution, use the following formula:

Note: if σ is known and the target population has a relative frequency distribution that is approximately normal, you can use the z-statistic to form the confidence interval.

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