Answer:
95% CI [20.823; 23.176]hours
Step-by-step explanation:
Hello!.
The study variable is:
X: Time that takes an architect to design a house. (Hours)
Sample n= 38 architects
Sample mean X[bar]= 22 hours
Sample Standard deviation S= 3.70 hours.
You are asked to make a 95% Confidence interval for the population mean. To study the population mean (μ) you need the variable to have a normal distribution. If the variable has a normal distribution, considering that the sample mean is a random variable derived from the random variable under study, then whatever the distribution of the variable under study will be, the distribution of the sample mean. Unfortunately, we don't have information about the distribution of the study variable.
Reminder - Central Limit Theorem definition
Be a population with probability function f(X;μ,δ²) from which a random sample of size n is selected. Then the distribution of the sample mean tends to the normal distribution with mean μ and variance δ²/n when the sample size tends to infinity.
As a rule, a sample of size greater than or equal to 30 is considered sufficient to apply the theorem and use the approximation.
So even if the study variable doesn't have normal distribution, applying the Theorem we can asume that X[bar]≈N(μ;δ²/n)
Now that the requirement of normal distribution has been met, we can use the standardization to construct the Confidence Interval:
[X[bar] ± [tex]Z_{1-\alpha /2}[/tex] * [tex]\frac{S}{\sqrt{n} }[/tex]]
[tex]Z_{1-\alpha /2} = Z_{0.975} = 1.96[/tex]
[22 ± 1.96 * [tex]\frac{3.70}{\sqrt{38} }[/tex]]
[20.823; 23.176]hours
With aconfidence level of 95% youd expect that the interval [20.823; 23.176]hours will contain the true value of the average time it takes architects to design a house.
I hope it helps!
