Question

1. the expected daily high temperature in tucson in november is 72 degrees with standard deviation of 8. x = the temperature on a randomly selected day. find:

a. p(x < 60)
b. p(x > 78)
c. p(70 < x < 79)
d. p(65 < x < 72)
e. p(x > 70)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 72\) (mean) and \(\sigma=8\) (standard deviation).

Part A

For \(x = 60\), \(z=\frac{60 - 72}{8}=\frac{-12}{8}=-1.5\)
\(P(X\lt60)=P(Z\lt - 1.5)\)
Using the standard normal table, \(P(Z\lt - 1.5)=0.0668\)

Part B

For \(x = 78\), \(z=\frac{78 - 72}{8}=\frac{6}{8}=0.75\)
\(P(X\gt78)=P(Z\gt0.75)\)
Since \(P(Z\gt z)=1 - P(Z\lt z)\), \(P(Z\gt0.75)=1 - 0.7734 = 0.2266\)

Part C

For \(x = 70\), \(z=\frac{70 - 72}{8}=\frac{-2}{8}=-0.25\)
For \(x = 79\), \(z=\frac{79 - 72}{8}=\frac{7}{8}=0.875\)
\(P(70\lt X\lt79)=P(-0.25\lt Z\lt0.875)\)
\(P(-0.25\lt Z\lt0.875)=P(Z\lt0.875)-P(Z\lt - 0.25)\)
\(P(Z\lt0.875)=0.8092\), \(P(Z\lt - 0.25)=0.4013\)
\(P(-0.25\lt Z\lt0.875)=0.8092 - 0.4013=0.4079\)

Part D

For \(x = 65\), \(z=\frac{65 - 72}{8}=\frac{-7}{8}=-0.875\)
For \(x = 72\), \(z=\frac{72 - 72}{8}=0\)
\(P(65\lt X\lt72)=P(-0.875\lt Z\lt0)\)
\(P(-0.875\lt Z\lt0)=P(Z\lt0)-P(Z\lt - 0.875)\)
\(P(Z\lt0)=0.5\), \(P(Z\lt - 0.875)=0.1908\)
\(P(-0.875\lt Z\lt0)=0.5 - 0.1908 = 0.3092\)

Part E

For \(x = 70\), \(z=\frac{70 - 72}{8}=-0.25\)
\(P(X\gt70)=P(Z\gt - 0.25)\)
Since \(P(Z\gt - z)=1 - P(Z\lt - z)\), \(P(Z\gt - 0.25)=1 - 0.4013=0.5987\)

Answer:

A. \(0.0668\)
B. \(0.2266\)
C. \(0.4079\)
D. \(0.3092\)
E. \(0.5987\)