Question

introduction to random variables and probability distributions and mean and standard deviation
random variables
calculating probabilities
x = the number of degrees a thermostat is set below the recommended 78 degrees

1. what is the probability that a randomly selected thermostat is set at 77 degrees?
2. what is the probability that a randomly selected thermostat is set less than 74 degrees?
3. what is the probability that a randomly selected thermostat is set for at least 75 degrees?
4. what is the probability that a randomly selected thermostat is set at 70 degrees?

Explanation:

Step1: Analyze question 1

The thermostat is set at 77 degrees. Since \(X\) is the number of degrees below 78, when \(X = 1\) (because \(78-77 = 1\)), from the table \(P(X = 1)=0.03\)

Step2: Analyze question 2

Less than 74 degrees. If \(X\) is the number of degrees below 78, for temperature less than 74 (\(78 - x<74\Rightarrow x > 4\) is wrong, correct: \(78 - X<74\Rightarrow X>4\) is wrong. Wait, no: \(X\) is degrees below 78. So for temperature \(T=78 - X\). We want \(T<74\), so \(78 - X<74\Rightarrow X > 4\). \(P(X>4)=P(X = 5)+P(X=6)=0.28 + 0.12=0.4\)

Step3: Analyze question 3

At least 75 degrees. \(T = 78 - X\geq75\Rightarrow X\leq3\). \(P(X\leq3)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)=0.03+0.03 + 0.05+0.18=0.29\)

Step4: Analyze question 4

Set at 70 degrees. \(T=70\), then \(X=78 - 70=8\). But \(X\) values in the table are \(0,1,\cdots,6\). So \(P(X = 8)=0\)

Answer:

1. \(0.03\)
2. \(0.4\)
3. \(0.29\)
4. \(0\)