Inference for Numerical Data

Car insurance savings. A market researcher wants to evaluate car insurance savings at a competing company. Based on past studies he is assuming that the standard deviation of savings is $100. He wants to collect data such that he can get a margin of error of no more than $10 at a 95% confidence level. How large of a sample should he collect?

Goal:

• the minimum sample size

Requirements:

• confidence level : 95%
• maximum allowable margin of error : 10$

\( mn ± Z^* * sd/sqrt(n) \)

• mn = mean
• sd = standard deviation
• n = sample size

\( Z^* * sd/sqrt(n) ≤ 10 \) { margin of error of no more than 10$ }

qnorm(0.025, mean=0, sd=1, lower.tail=F)

[1] 1.959964

• Z* = 1.96
• sd = 100

Substituting the values in the formula:

\( 1.96*100/sqrt(n) ≤ 10 \)

\( = 19.6 ≤ sqrt(n) \)

\( = 19.6^2 ≤ n \)

\( = 384.16 ≤ n \)

\( n = 385 \)

n<- 19.6^2
n 

[1] 384.16

n<- 19.6^2
n 

[1] 384.16

Therefore, the minimum sample size to have a margin of error of $10 is 385 observations.