The owner of a coffee shop has found that the average amount spent by 25 customers at the shop is...

Question:

The owner of a coffee shop has found that the average amount spent by 25 customers at the shop is $9.70. Based on a large number of similar studies, it is known that the variance for the amount spent by all customers is$1.96.

What is the 90% confidence interval for the population mean amount spent by customers?

Confidence Interval:

In the probability and statistics theory, the confidence interval of the population parameter contains the values for the estimate of population parameter (known as statistic based on sample taken from the population).

Given Information:

The owner of a coffee shop has found that the average amount spent by 25 customers at the shop is 9.70. Based on a large number of similar studies, it is known that the variance for the amount spent by all customers is 1.96.

The sample size is {eq}n = 25. {/eq}

The sample mean is {eq}\bar x = 9.7 {/eq}

The population standard deviation is {eq}\sigma = 1.96 {/eq}

Since, the sample size is not large enough but the population variance is known, hence the confidence interval is computed by using the normal distribution.

The formula for confidence interval is given as,

{eq}\bar x \pm {z_{\dfrac{\alpha }{2}}} \cdot \dfrac{\sigma }{{\sqrt n }} {/eq}

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

And, {eq}{z_{\dfrac{\alpha }{2}}} {/eq} is the standard normal critical value at a level of significance {eq}\alpha {/eq}.

The lower limit of 90% confidence interval is:

{eq}\begin{align*} \bar x - {z_{0.025}} \cdot \dfrac{\sigma }{{\sqrt n }} &= 2.7 - \left( {1.6449} \right) \cdot \dfrac{{1.96}}{{\sqrt {25} }}\\ &= 2.055 \end{align*} {/eq}

The upper limit of 90% confidence interval is:

{eq}\begin{align*} \bar x + {z_{0.025}} \cdot \dfrac{\sigma }{{\sqrt n }} &= 2.7 + \left( {1.6449} \right) \cdot \dfrac{{1.96}}{{\sqrt {25} }}\\ &= 3.345 \end{align*} {/eq}

The z-value is obtained using Excel software.

Therefore, 90% confidence interval for the population mean amount spent by customers is {eq}\left( {2.055,3.345} \right). {/eq}