Basics
What is a confidence interval used for?
To estimate a range where a population parameter is likely to be
Which distribution (Z or T) is used when the sample size is large?
Z
What Excel function calculates the standard error if
σ = 8 and n = 36?
= 8 / SQRT(36)
Find the Z-score for a 99% confidence level
= - NORM.S.INV(0.01/2) or
= - NORM.S.INV(0.005)
= 2.576
A sample of n = 22 has a sample standard deviation. Find the T-critical value for a 90% confidence level
= - T.INV(0.1/2, 22-1) or
= - T.INV(0.05, 21)
= 1.721
What happens to the confidence interval when we increase the confidence level?
The interval gets wider because we want to be more sure the true value is inside
What is the degrees of freedom (df) when using the T-distribution for a sample of size 40?
df = n - 1 = 40 - 1 = 39
What Excel formula finds the Z-score for a 95% confidence level?
= - NORM.S.INV(0.05/2) or
= - NORM.S.INV(0.025)
A sample of n = 25 has a 95% confidence level. Find the critical value
= - T.INV(0.05/2, 25-1) or
= - T.INV(0.025, 24)
= 2.064
Which distribution (Z or T) has wider tails and why?
T-distribution, because it accounts for extra uncertainty in estimating σ
Why does a larger sample size create a narrower confidence interval?
A larger sample reduces variability, making the estimate more precise
As the sample size increases, what happens to the shape of the T-distribution?
It approaches the normal (Z) distribution because the sample standard deviation (s) becomes a better estimate of the population standard deviation (σ)
What Excel formula finds the T-score for a 99% confidence level
with n = 20?
= - T.INV(0.01/2, 20-1) or
= - T.INV(0.005, 19)
Find the 90% confidence interval for a sample with
mean = 50, σ = 10, n = 36
47.26 to 52.74
What Excel function finds the margin of error for p̄ = 0.5 and n=200 at 95% confidence?
= - NORM.S.INV(0.05/2) *
SQRT(0.5 * (1 - 0.5) / 200)
If a 95% confidence interval for the average GPA of CSULB students is 2.8 to 3.4, what does this mean?
We are 95% confident that the true population mean of CSULB students' GPA lies between 2.8 and 3.4
A student calculates a 95% confidence interval using Z when they should use T. How does this mistake affect their result?
The interval will be too narrow, underestimating uncertainty
What does the
= - T.INV(0.05, 9)
function return?
The T-critical value for a 90% confidence level and df = 9
Find the 95% confidence interval for p̄ = 0.7, n = 150
62.67% to 77.33%
What function finds the T-score for a 99% confidence level with n = 25?
= - T.INV(0.01/2, 25-1) or
= - T.INV(0.005, 24)
What two things affect the width of a confidence interval?
Sample size and
confidence level
If you don’t know the population standard deviation and have a small sample size, do you use Z or T?
T
If the population standard deviation is known, what Excel function helps find the Z-critical value for a 98% confidence level?
= - NORM.S.INV(0.02/2) or
= - NORM.S.INV(0.01)
Find the T-critical value for a 98% confidence level
with n = 15
2.624
Why does a 99% confidence interval have a larger margin of error than a 90% confidence interval?
Because a higher confidence level requires a wider range to ensure the true value is captured