# For a dataset of size 58, the sample mean was calculated to be #barx=$2.75# and the sample standard deviation is #s=$0.86#. What is the 95% confidence interval for #mu#?

##### 1 Answer

The 95% confidence interval for

#### Explanation:

The formula for a 95% confidence interval for

#barx+-[t_(alpha//2, n-1)xxs/sqrtn]#

where

#bar x# is your sample mean, the middle point of the confidence interval,#t_(alpha//2, n-1)# is a stretching factor that tells us how many 'standard errors' wide our interval needs to be,#s# is the standard deviation of the sample data points, and#n# is the number of data points, also called the sample size.

The term **standard error** for the estimate of **standard deviation** of the whole sample. (More on that later.)

To find the 95% confidence interval, we just need to plug in the given values for the variables, and look up the

#color(white)= barx+-[t_(alpha//2, n-1)xxs/sqrtn]#

#=2.75+-[t_(0.05//2, 58-1)xx0.86/sqrt58]#

#=2.75+-[t_(0.025, 57)xx0.1129]#

#=2.75+-[2.002465xx0.1129]#

#=2.75+-0.2261#

#=(2.5239," "2.9761)#

## Bonus:

**Standard deviation** measures the spread of all the data as a whole. **Standard error** measures how close to the actual population mean *the data as a whole* less and less, allowing

Also: for high enough values of

for sufficiently large

#n# (greater than 30, usually), it is common to approximate#t_(alpha//2, n-1)# as#z_(alpha//2)# .

It is much easier to look up