Question

How do I construct a $95\%$ confidence interval for the average amount spent on lunch by all customers?

Answer 1

Hint: To solve this question we need to know the concept of confidence interval. To solve the problem we need to calculate the following steps: First calculate the significance level, the second step is the calculation critical value, the third step is to find the margin of error and then to find the lower and the upper limit, thus finding the interval.

Complete step-by-step answer:
The question asks us to find a $95\%$ confidence interval for the average amount spent on the lunch given by all the customers. To find the confidence interval we need to consider the following steps. The first step is to find the significance level, which is denoted by $\alpha $. So the formula for $\alpha $ is the difference between $1$ and confidence which mathematically is denoted by:
$\text{Significance level = }\!\!\alpha\!\!\text{ = 1- confidence}$
Confidence in the percentage format is given in the question that is $95\%$ which when converted into decimal becomes $0.95$ . The second step for solving the problem is to find the critical value. The value of critical value depends upon the significance level, which means for different values of significance level which is $\alpha $ we get different critical values which we get from the z-table.
The third step in the process is to find the margin of error is the product of critical value and standard error which is given in the question.
After getting the value of margin of error we are supposed to find the limits of $95\%$ of confidence interval. To find this we will have to get the lower limit and the upper limit which is denoted by the sample mean minus the margin of error and the sample mean plus the margin of error, which is mathematically written as
$\text{Lower limit = Sample Mean - Margin Of Error}$
$\text{Upper limit = Sample Mean + Margin Of Error}$
The confidence interval thus becomes from lower limit to upper limit.

Note: To convert the percentage into decimal we need to change the percentage into fraction and then to convert the fraction into decimal. To convert the percentage to fraction the fraction is divided by 100 and the percentage sign is removed from the number. For instance, let a percentage be “a%” which on changing into fraction becomes $\dfrac{a}{100}$ which on further conversion into decimal becomes $0.0a$ .