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Answer:
A) 0.5000; B) 0.9772; C) 0.9772; D) 0.8185; E) 0.9932; F) 1.0000
Step-by-step explanation:
For each of these, we will use a z score. The formula for a z score is:
A) Our X is 5, our mean is 5 and our standard deviation is 0.2:
z = (5-5)/(0.2) = 0/0.2 = 0
Using a z table, the value to the left of this, less than, is 0.5000; this means the area to the right of this, greater than, is 1-0.5000 = 0.5000.
B) Our X is 5.4, the mean is 5 and the standard deviation is 0.2:
z = (5.4-5)/0.2 = 0.4/0.2 = 2
Using a z table, the value to the left of this, less than, is 0.9772.
C) The probability for this will be the same as for B; there is no distinction between "less than" and "less than or equal to" in z tables.
D) We find the z score for each of the endpoints of this interval, find the probabilities and subtract them:
z = (4.6-5)/0.2 = -0.4/0.2 = -2; the probability is 0.0228.
z = (5.2-5)/0.2 = 0.2/0.2 = 1; the probability is 0.8413.
The area between them is 0.8413-0.0228 = 0.8185.
E) X is 4.5, the mean is 5 and the standard deviation is 0.2:
z = (4.5-5)/0.2 = -0.5/0.2 = -2.5; the probability less than this is 0.0062. This means the probability greater than this is 1-0.0062 = 0.9938.
F) X is 4, the mean is 5 and the standard deviation is 0.2:
z = (4-5)/0.2 = -1/0.2 = -5. Everything in the z table is larger than this, so the probability is 1.000.