# Calculate a 95% Confidence Interval for the following samples: bar{X} = 19, N = 17, s = 5

## Question:

Calculate a 95% Confidence Interval for the following samples:

{eq}\bar{X} = 19, N = 17, s = 5 {/eq}

## Confidence Interval:

The confidence interval is used to estimate true value of population parameter. The Student's t-distribution is used when the sample is small, and the population variance is not known. The formula for confidence interval is given as,

{eq}\bar x \pm {t_{\alpha /2}} \cdot \dfrac{s}{{\sqrt n }} {/eq}

Here,

{eq}\bar x {/eq} is sample mean.

{eq}{t_{\alpha /2}} {/eq} is the critical value at level of significance {eq}\alpha {/eq} and degree of freedom {eq}n - 1. {/eq}

Given Information:

The sample size is {eq}N = 17. {/eq}

The sample mean is {eq}\bar X = 19. {/eq}

The sample standard deviation is {eq}s = 5. {/eq}

The lower limit of 95% confidence interval is given as,

{eq}\begin{align*} \bar x - {t_{0.025,16}} \cdot \dfrac{s}{{\sqrt n }} &= 19 - \left( {2.473} \right) \cdot \dfrac{5}{{\sqrt {17} }}\\ &= 16.001 \end{align*} {/eq}

The upper limit of 95% confidence interval is given as,

{eq}\begin{align*} \bar x + {t_{0.025,16}} \cdot \dfrac{s}{{\sqrt n }} &= 19 + \left( {2.473} \right) \cdot \dfrac{5}{{\sqrt {17} }}\\ &= 21.999 \end{align*} {/eq}

Therefore, the 95% confidence interval is {eq}\left( {16.001,\,21.999} \right). {/eq}