1-pnorm(1.2)
## [1] 0.1150697
pnorm(0.53)
## [1] 0.701944
  • to find areas using a table: find your z-score on the header and first column of the table. The outside (axis) of the table
  • to find a z-score corresponding to an area: find the area inside the table (in the main part of the table) and then read off the z-score from the outside.
  1. (Using a table) A population of early career professionals has an average salary of \(\$50,000\) with a standard deviation of \(\$15,000\). Find a range of salaries for which the average salary of a sample of \(200\) professionals will be in the range with \(95 \%\) probability.
se <- 15000/sqrt(200) 
qnorm(.025)*se + 50000
## [1] 47921.14
qnorm(.975)*se + 50000
## [1] 52078.86
# 47921.14 to 52078.86
# From a sample of 200 professionals with population mean 50,000 and standard deviation 15,000, there is a 95% chance that the sample mean will be between 47921.14 and 52078.86
  1. (Using a table) A population has mean \(\mu = 143\) and standard deviation \(\sigma = 15\). Describe the sampling distribution for a sample of size \(150\). Draw a picture of the sampling distribution. Label the area corresponding to the probability of a sample mean greater than \(144\). Find the probability.

CLT for Proportions

\[ \hat{p} \sim N\left(p,\sqrt{\frac{p(1-p)}{n}}\right)\]

  1. (Using a table) The true proportion of red skittles made at the factory is \(35 \%\). From a sample of \(100\) skittles, what is the probability that the proportion of skittles will be greater than \(40 \%\)?

  2. (Using a table) The true proportion of red skittles made at the factory is \(35 \%\). From a sample of \(100\) skittles, you are really hoping to get an unusually large number of red skittles. What proportion of red skittles would be in the top \(1 \%\) of samples.

  3. Find the middle \(95 \%\)

Population Parameters aren’t known

\[ \mu - 1.96 \cdot SE \leq \bar{X} \leq \mu + 1.96 \cdot SE \]