Question

1. Calculate the missing value for the probability P(Z <?)= 0.5. (1) 1 (2) -1 (3) 0 (4) 0.5 (5) -0.5
2. Consider a population with a mean of 70 and a standard deviation of 6. A random sample of size 36 is drawn. What is the probability that the sample mean is between 70.5 and 71.5? (1) 0.9332 (2) 0.5000 (3) 0.6915 (4) 0.2415 (5) None of the above
3. Consider the information provided in question 2. What is the probability that the sample mean is at least 67.5? (1) 2.5 (2) 0.9938 (3) 0.0062 (4) -2.5 (5) None of the above

Answer 1

1)P(Z <?)= 0.5.

The midpoint of a normal distribution curve is at Z=0, the area to the left is equal to the area to the right. 0.5.

The answer is (3) 0

2)P(70.5 <$\bar x$ <71.5)

Mean =70, SD =6, n=36

Z=$\frac {(\bar x - mean) \sqrt n} {SD}$

Z1=$\frac {(70.5-70) \sqrt{36}}{6}$

Z1=0.5

Z2=$\frac {(71.5-70) \sqrt {36}}{6}$

Z2=1.5

=$\Phi(1.5)-\Phi(0.5)$

The values are read from a Z-table

=0.93319-0.69146

=0.24173

(4)0.2415

3)P($\bar x \geq 67.5)$

Z=$\frac {(67.5-70)\sqrt {36}} {6}$

Z=-2.5

$\Phi(-2.5)$

=0.9938

(2)0.9938