Question

assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of (0^{circ}c) and a standard deviation of (1.00^{circ}c). a single thermometer is randomly selected and tested. find (p_{22}), the 22 - percentile. this is the temperature reading separating the bottom 22% from the top 78%. (p_{22}=) (^{circ}c) (round answer to three decimal places) question help: video

Explanation:

Step1: Use the z - score formula for percentile

We know that for a normal distribution \(X\sim N(\mu,\sigma^{2})\), and we want to find the \(z\) - score \(z\) such that \(P(X < z)=\alpha\). Here \(\mu = 0\), \(\sigma=1\) (standard normal distribution), and \(\alpha = 0.22\).
We can use a standard normal table (or a calculator with a normal - distribution function, such as the inverse of the cumulative distribution function for the standard normal distribution, denoted as \(z=\Phi^{-1}(0.22)\)).

Step2: Find the z - score

Using a calculator (e.g., in R: qnorm(0.22) or in Python scipy.stats.norm.ppf(0.22)), or a standard normal table (looking for the value closest to \(0.22\) in the body of the table).
The \(z\) - score corresponding to a cumulative probability of \(0.22\) is approximately \(z=- 0.772\)

Answer:

\(-0.772\)