Confidence intervals, P values, Error and Power

Question 1

What is the purpose of confidence intervals?

Answer 1

When we take a sample, the mean is an estimation of the population mean but we have some randomness. Confidence intervals are an attempt to quantify that randomness, giving us a range via an upper and lower bound in which If we would repeat the experiment, in __% of the time, the true pop. mean ( ) will fall within the constructed interval.

For example the classic 95% confidence interval:
95%CI : P(x −1.96 ∗SE ≤μ≤+ x + 1.96 ∗SE)

α= 0.05
left border: x ̄ −1.96 ∗SE
right border: x ̄ + 1.96 ∗SE

Question 2

Where does this value of 1.96 come from?

Answer 2

1.96 standard deviations either direction of a normal distribution constitutes a range which contains 95% of the values in the distribution

Question 3

How can you show this (1.96) using R functions? (2)

Answer 3

qnorm(.025, 0, 1) # left border in standard normal distribution
## [1] -1.959964

pnorm(-1.96)
## [1] 0.0249979

Question 4

If we simulated data with 100 samples of 30 and 100 samples of 300:

Would there be less CI that don’t include the true mean?

Does the width of the CI change?

Answer 4

since sd/sqrt(N) is affected by the sample mean in:
P(x −1.96 ∗ sd/sqrt(N) ≤μ≤+ x + 1.96 ∗ sd/sqrt(N))
the width of the CI does change and become smaller with higher numbers.

However we are still collecting 95% confidence intervals so the frequency does not change. Instead the CI’s become more informative as the range is smaller