Question

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`?

Answer 1

The 95% confidence interval for `mu` is `($2.5239,"  "$2.9761)`.

Explanation:

The formula for a 95% confidence interval for `mu` is

`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.

`alpha` is just 100% minus the confidence level you wish to have (in this case 95%). So `alpha = 100%-95%`, which is 5%, or `0.05`.

The term `s/sqrtn` is known as the standard error for the estimate of `mu`. This is different than the 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 `t`-value in a table (or use computer software).

`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 `(mu)` our sample mean `(barx)` is. Each additional data point affects the spread of the data as a whole less and less, allowing `barx` to stabilize. In math terms: as `n` increases, `s` gets more precise, and `s/sqrtn` gets closer to 0.

Also: for high enough values of `n`, the `t`-distribution can be approximated by a normal distribution, allowing for easier table lookup. That is,

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 `z_(alpha//2)` than it is to look up `t_(alpha//2, n-1)`.