To calculate the lower limit of an approximate 95% confidence interval for the population average number of cups of coffee, you can use the formula:
Lower limit = sample mean - (critical value * standard error)
First, calculate the standard error using the formula:
Standard error = population standard deviation / √(sample size)
In this case, the population standard deviation is given as 6 cups and the sample size is 40. Plugging these values in:
Standard error = 6 / √40 ≈ 0.9486833
Next, find the critical value corresponding to a 95% confidence level. This critical value depends on the sample size and is obtained from a t-distribution table. Since your sample size is 40, you will use a t-distribution with 39 degrees of freedom. The critical value for a 95% confidence level with 39 degrees of freedom is approximately 2.022.
Now you can calculate the lower limit:
Lower limit = 20 - (2.022 * 0.9486833) ≈ 20 - 1.920651 ≈ 18.079
Rounding to three decimal places, the lower limit of an approximate 95% confidence interval for the population average cup of coffee is approximately 18.079.
The closest option is (4) 18.141, so your calculation is correct!