Question

1. reaction times have μ = 250 ms and σ = 40 ms. find the probability that a reaction time is between 220 ms and 295 ms.
2. daily water consumption per household is normally distributed with μ = 250 gallons and σ = 30 gallons. find the probability that a household uses less than 235 gallons.
3. battery lifespans follow a normal distribution with μ = 500 days and σ = 80 days. find the probability that a battery lasts more than 580 days.
4. sat math scores are normally distributed with μ = 520 and σ = 100. find the probability that a student scores between 540 and 600.
5. resting heart rates for adults are normally distributed with μ = 72 bpm and σ = 8 bpm. find the probability that a persons heart rate is below 66 bpm.

Explanation:

Step1: Calculate the z - score for \(x = 220\)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 250\), \(\sigma=40\), and \(x = 220\), we have \(z_1=\frac{220 - 250}{40}=\frac{- 30}{40}=-0.75\)

Step2: Calculate the z - score for \(x = 295\)

Using the same formula \(z=\frac{x-\mu}{\sigma}\), with \(x = 295\), \(\mu = 250\), and \(\sigma = 40\), we get \(z_2=\frac{295-250}{40}=\frac{45}{40}=1.125\)

Step3: Find the probabilities corresponding to the z - scores

We use the standard normal distribution table (or a calculator with a normal - distribution function). The probability \(P(Z\lt z_1)\) where \(z_1=-0.75\) is \(P(Z\lt - 0.75)=0.2266\) (from the standard normal table). The probability \(P(Z\lt z_2)\) where \(z_2 = 1.125\) is \(P(Z\lt1.125)=0.8697\) (using a calculator or more detailed table).

Step4: Calculate the probability between the two z - scores

The probability \(P(220\lt X\lt295)=P(Z\lt1.125)-P(Z\lt - 0.75)\)
Substitute the values: \(P(220\lt X\lt295)=0.8697 - 0.2266=0.6431\)

Answer:

The probability that a reaction time is between \(220\) ms and \(295\) ms is \(0.6431\)