Working backwards, Part II

A 90% confidence interval for a population mean is (65, 77). The population distribution is approximately normal and the population standard deviation is unknown. This confidence interval is based on a simple random sample of 25 observations. Calculate the sample mean,the margin of error, and the sample standard deviation.

Sample Mean

sample mean is (x2+x1) / 2 where the confidence interval is (x1, x2)

n <- 25
x1 <- 65
x2 <- 77

SM <- (x2 + x1) / 2
cat("The Sample Mean is", SM)
## The Sample Mean is 71

Margin of Error

Margin of error is (x2−x1) / 2 where the confidence interval is (x1, x2)

n <- 25
x1 <- 65
x2 <- 77

ME <- (x2 - x1) / 2
cat("The Margin of error is", ME)
## The Margin of error is 6

Sample standard deviation

sample standard devation using ME = t(.05)*s/sqrt(n). Using the qt function and df = 25-1

df <- 25-1
tval <- qt(.95, df)
sd <- (ME/tval)*5
sd
## [1] 17.53481

SAT scores.

SAT scores of students at an Ivy League college are distributed with a standard deviation of 250 points. Two statistics students, Raina and Luke, want to estimate the average SAT score of students at this college as part of a class project. They want their margin of error to be no more than 25 points.

(a) Raina wants to use a 90% confidence interval. How large a sample should she collect?

25 = z ∗ 250 / √n

or n=(10∗z)2

# estimate z score of 90% confidence interval is 1.645
z <- 1.645 
margin <- 25
stand_dev <- 250
z90Raina <- round(((stand_dev/margin) * z)^2, 0)
z90Raina
## [1] 271

(b) Luke wants to use a 99% confidence interval. Without calculating the actual sample size, determine whether his sample should be larger or smaller than Raina’s, and explain your reasoning.

## Luke’s sample should be larger because it requires a higher z number multiplied by the standard deviation and then squared.

(c) Calculate the minimum required sample size for Luke.

# estimate z score of 99% confidence interval is 2.576
z <- 2.576
margin <- 25
stand_dev <- 250

z99Luke <- round(((stand_dev/margin) * z)^2, 0)
z99Luke
## [1] 664