The following random sample from a population whose values were normally distributed was collected.10 8 11 11
The 95% confidence interval for $\mu$ is
A)8.52 to 10.98
B)7.75 to 12.25
C)9.75 to 10.75
D)8.00 to 10.00

Question

Quizplus · Accepted Answer

The formula for calculating the 95% confidence interval for the population mean (μ) is:

CI = x̄ ± z*(σ/√n)

Where:
x̄ = sample mean
z = z-score for desired confidence level (1.96 for 95%)
σ = population standard deviation (unknown, so we estimate it with the sample standard deviation)
n = sample size

Plugging in the given values, we get:

CI = 10 ± 1.96*(s/√4)

Where s is the sample standard deviation.

To find s, we first need to calculate the sample variance:

s^2 = Σ(xi - x̄)^2 / (n - 1)

= [(10-10.0)^2 + (8-10.0)^2 + (11-10.0)^2 + (11-10.0)^2] / (4-1)

= 5.67

Then, we can take the square root to get s:

s = √5.67

= 2.38

Plugging this into the CI formula, we get:

CI = 10 ± 1.96*(2.38/√4)

= 10 ± 1.96*1.19

= 10 ± 2.33

= [7.67, 12.33]

So the best choice is B, 7.75 to 12.25, which is the closest option to the calculated interval.

The Following Random Sample from a Population Whose Values Were μ\muμ

The Following Random Sample from a Population Whose Values Were $\mu$