The mean time taken to design a house plan by 38 architects was found to be 22 hours with a standard deviation of 3.70 hours. a. construct a 95% confidence interval for the population mean μ. round your answers to two decimal places.

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] ± * ]

[22 ± 1.96 * ]

[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!

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step-by-step explanation:

first one because it can't have a fraction in it

step-by-step explanation: