7.1 Sampling Plans and Experimental Designs

7.2 Statistics and Sampling Distributions

7.3 The Central Limit Theorem and the Sample Mean

Example 7.5 To determine whether a bottling machine is working satisfactorily, a production line manager randomly samples ten 12-ounce bottles every hour and measures the amount of beverage in each bottle. The mean x of the 10 fill measurements is used to decide whether to readjust the amount of beverage delivered per bottle by the filling machine. If records show that the amount of fill per bottle is normally distributed, with a standard deviation of .2 ounce, and if the bottling machine is set to produce a mean fill per bottle of 12.1 ounces, what is the approximate probability that the sample mean x of the 10 test bottles is less than 12 ounces?

u <- 12.1
sigma <- .2
n <- 10
ee <- sigma/sqrt(10)
pnorm(12, 12.1, ee)
## [1] 0.05692315
  1. A random sample of size \(n=49\) is selected from a population with mean \(\mu=53\) and standard deviation \(\sigma=21\) find the probability that the sample mean is greater than 55.
u <- 53
sigma <- 21
n <- 49
ee <- sigma/sqrt(n)
pnorm(55, u, ee, lower.tail = FALSE)
## [1] 0.2524925
  1. A random sample of size \(n=40\) is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=20\), find the probability that the sample mean is between 105 and 110
u <- 100
sigma <- 20
n <- 40
ee <- sigma/sqrt(n)
pnorm(110, u, ee)-pnorm(105, u, ee)
## [1] 0.05614045

20.A random sample of size \(n=25\) is selected from a normal population with mean \(\mu=106\) and standard deviation \(\sigma=12\),

  1. Find the probability that the sample mean exceeds 110
  2. Find the probability that the sample mean deviates from the population mean \(\mu=106\) by no more than 4
u <- 106
sigma <- 12
n <- 25
ee <- sigma/sqrt(n)
pnorm(110, u, ee, lower.tail = FALSE)
## [1] 0.04779035
pnorm(110, u, ee, lower.tail = TRUE)-pnorm(102, u, ee, lower.tail = TRUE)
## [1] 0.9044193